Josafhat Salinas Ruíz
Osval Antonio Montesinos López
Gabriela Hernández Ramírez
Jose Crossa Hiriart
Generalized Linear
Mixed Models
with Applications
in Agriculture and
Biology
Generalized Linear Mixed Models with Applications
in Agriculture and Biology
Josafhat Salinas Ruíz • Osval Antonio Montesinos
López • Gabriela Hernández Ramírez
Jose Crossa Hiriart
Generalized Linear Mixed
Models with Applications
in Agriculture and Biology
Josafhat Salinas Ruíz
Innovación Agroalimentaria Sustentable
Colegio de Postgraduados
Córdoba, Mexico
Osval Antonio Montesinos López
Facultad de Telemática
University of Colima
Colima, Mexico
Gabriela Hernández Ramírez
Ciencia de los Alimentos y Biotecnología
Tecnológico Nacional de México/ITS
de Tierra Blanca, Mexico
Jose Crossa Hiriart
International Maize and Wheat Improvement
Center (CIMMYT)
Texcoco, Mexico
ISBN 978-3-031-32799-5
ISBN 978-3-031-32800-8
https://doi.org/10.1007/978-3-031-32800-8
(eBook)
The translation was done with the help of artificial intelligence (machine translation by the service
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© The Editor(s) (if applicable) and The Author(s) 2023. This book is an open access publication.
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Foreword
This book is another example of CIMMYT’s commitment to scientific advancement,
and comprehensive and inclusive knowledge sharing and dissemination. By using
alternative statistical models and methods to describe and analyze data sets from
different disciplines, such as biology and agriculture, the book facilitates the adoption and effective use of these tools by publicly funded researchers and practitioners
of national agricultural research extension systems (NARES) and universities across
the Global South.
The authors aim to offer different and new models, methods, and techniques to
agricultural scientists who often lack the resources to adopt these tools or face
practical constraints when analyzing different types of data.
This work would not be possible with the continuous support of CIMMYT’s
outstanding partners and donors who invest in non-profit frontier research for the
benefit of millions of farmers and low-income communities worldwide. For that
reason, it could not be more fitting for this book to be published as an open access
resource for the international community to benefit from. I trust that this publication
will greatly contribute to accelerate the development and deployment of resourceefficient and nutritious crops for a food secure future.
Bram Govaerts
Director General, CIMMYT
v
Acknowledgments
The work presented in this book described models and methods for improving the
statistical analyses of continuous, ordinal, and count data usually collected in
agriculture and biology. The authors are grateful to the past and present CIMMYT
Directors General, Deputy Directors of Research, Deputy Directors of Administration, Directors of Research Programs, and other Administration offices and Laboratories of CIMMYT for their continuous and firm support of biometrical genetics, and
statistics research, training, and service in support of CIMMYT’s mission: “maize
and wheat science for improved livelihoods.”
This work was made possible with support from the CGIAR Research Programs
on Wheat and Maize (wheat.org, maize.org), and many funders including Australia,
United Kingdom (DFID), USA (USAID), South Africa, China, Mexico
(SAGARPA), Canada, India, Korea, Norway, Switzerland, France, Japan, New
Zealand, Sweden, and the World Bank. We thank the financial support of the Mexico
Government throughout MASAGRO and several other regional projects and close
collaboration with numerous Mexican researchers.
We acknowledge the financial support provided by the (1) Bill and Melinda Gates
Foundation (INV-003439 BMGF/FCDO Accelerating Genetic Gains in Maize and
Wheat for Improved Livelihoods [AG2MW]) as well as (2) USAID projects
(Amend. No. 9 MTO 069033, USAID-CIMMYT Wheat/AGGMW, AGG-Maize
Supplementary Project, AGG [Stress Tolerant Maize for Africa]).
Very special recognition is given to Bill and Melinda Gates Foundation for
providing the Open Access fee of this book.
We are also thankful for the financial support provided by the (1) Foundations for
Research Levy on Agricultural Products (FFL) and the Agricultural Agreement
Research Fund (JA) in Norway through NFR grant 267806, (2) Sveriges
Llantbruksuniversitet (Swedish University of Agricultural Sciences) Department of
The original version of this book has been revised. The Acknowledgment section which was
inadvertently omitted after the Foreword has now been included.
vii
viii
Acknowledgments
Plant Breeding, Sundsvägen 10, 23053 Alnarp, Sweden, (3) CIMMYT CRP, (4) the
Consejo Nacional de Tecnología y Ciencia (CONACYT) of México, and (5)
Universidad de Colima of Mexico.
We highly appreciate and thank the several students and professors at the
Universidad de Colima and students as well professors from the Colegio de PostGraduados (COLPOS) who tested and made suggestions on early version of the
material covered in the book; their many useful suggestions had a direct impact on
the current organization and the technical content of the book.
Contents
1
2
Elements of Generalized Linear Mixed Models . . . . . . . . . . . . . . . . .
1.1
Introduction to Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Simple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . . . .
1.3
Analysis of Variance Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 One-Way Analysis of Variance . . . . . . . . . . . . . . . . . . . .
1.3.2 Two-Way Nested Analysis of Variance . . . . . . . . . . . . . .
1.3.3 Two-Way Analysis of Variance with Interaction . . . . . . .
1.4
Analysis of Covariance (ANCOVA) . . . . . . . . . . . . . . . . . . . . .
1.5
Mixed Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Mixed Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.3 Distribution of the Response Variable Conditional
on Random Effects (y| b) . . . . . . . . . . . . . . . . . . . . . . . .
1.5.4 Types of Factors and Their Related Effects
on LMMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.5 Nested Versus Crossed Factors and Their
Corresponding Effects . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.6 Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.7 One-Way Random Effects Model . . . . . . . . . . . . . . . . . .
1.5.8 Analysis of Variance Model of a Randomized Block
Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Components of a GLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The Random Component . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
2
4
6
6
9
13
18
22
22
23
25
26
27
27
30
31
35
40
43
43
44
44
ix
x
3
4
Contents
2.2.2 The Systematic Component . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Predictor’s Link Function η . . . . . . . . . . . . . . . . . . . . . .
2.3
Assumptions of a GLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Estimation and Inference of a GLM . . . . . . . . . . . . . . . . . . . . . .
2.5
Specification of a GLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Continuous Normal Response Variable . . . . . . . . . . . . . .
2.5.2 Binary Logistic Regression . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Poisson Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.4 Gamma Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.5 Beta Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
45
48
48
49
49
51
59
67
72
76
83
Objectives of Inference for Stochastic Models . . . . . . . . . . . . . . . . . .
3.1
Three Aspects to Consider for an Inference . . . . . . . . . . . . . . . . .
3.1.1 Data Scale in the Modeling Process Versus
Original Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Inference Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Inference Based on Marginal and Conditional
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Illustrative Examples of the Data Scale and the Model Scale . . . .
3.3
Fixed and Random Effects in the Inference Space . . . . . . . . . . . .
3.3.1 A Broad Inference Space or a Population Inference . . . . .
3.3.2 Mixed Models with a Normal Response . . . . . . . . . . . . .
3.4
Marginal and Conditional Models . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Marginal Versus Conditional Models . . . . . . . . . . . . . . .
3.4.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Non-normal Distribution . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
86
Generalized Linear Mixed Models for Non-normal Responses . . . . .
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
A Brief Description of Linear Mixed Models (LMMs) . . . . . . . .
4.3
Generalized Linear Mixed Models . . . . . . . . . . . . . . . . . . . . . . .
4.4
The Inverse Link Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
The Variance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Specification of a GLMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7
Estimation of the Dispersion Parameter . . . . . . . . . . . . . . . . . . .
4.8
Estimation and Inference in Generalized Linear Mixed
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.2 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9
Fitting the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
87
88
88
101
101
103
105
105
107
108
111
113
113
114
114
116
117
117
119
123
123
125
126
126
Contents
5
6
Generalized Linear Mixed Models for Counts . . . . . . . . . . . . . . . . . .
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
The Poisson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 CRD with a Poisson Response . . . . . . . . . . . . . . . . . . . .
5.2.2 Example 2: CRDs with Poisson Response . . . . . . . . . . . .
5.2.3 Example 3: Control of Weeds in Cereal Crops
in an RCBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Overdispersion in Poisson Data . . . . . . . . . . . . . . . . . . . .
5.2.5 Factorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6 Latin Square (LS) Design . . . . . . . . . . . . . . . . . . . . . . . .
5.3
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Linear Mixed Models for Proportions
and Percentages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
Response Variables as Ratios and Percentages . . . . . . . . . . . . . .
6.2
Analysis of Discrete Proportions: Binary and Binomial
Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Completely Randomized Design (CRD): Methylation
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
Factorial Design in a Randomized Complete Block Design
(RCBD) with Binomial Data: Toxic Effect of Different
Treatments on Two Species of Fleas . . . . . . . . . . . . . . . . . . . . .
6.4
A Split-Plot Design in an RCBD with a Normal Response . . . . . .
6.4.1 An RCBD Split Plot with Binomial Data: Carrot Fly
Larval Infestation of Carrots . . . . . . . . . . . . . . . . . . . . . .
6.5
A Split-Split Plot in an RCBD:- In Vitro Germination
of Seeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6
Alternative Link Functions for Binomial Data . . . . . . . . . . . . . . .
6.6.1 Probit Link: A Split-Split Plot in an RCBD
with a Binomial Response . . . . . . . . . . . . . . . . . . . . . . .
6.6.2 Complementary Log-Log Link Function: A Split Plot
in an RCBD with a Binomial Response . . . . . . . . . . . . . .
6.7
Percentages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7.1 RCBD: Dead Aphid Rate . . . . . . . . . . . . . . . . . . . . . . . .
6.7.2 RCBD: Percentage of Quality Malt . . . . . . . . . . . . . . . . .
6.7.3 A Split Plot in an RCBD: Cockroach Mortality
(Blattella germanica) . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7.4 A Split-Plot Design in an RCBD: Percentage Disease
Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7.5 Randomized Complete Block Design with a Binomial
Response with Multiple Variance Components . . . . . . . .
6.8
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
129
129
129
131
134
138
142
149
157
170
178
209
209
210
210
217
219
221
232
237
238
239
242
242
245
247
250
253
257
265
xii
7
8
9
Contents
Time of Occurrence of an Event of Interest . . . . . . . . . . . . . . . . . . . .
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Generalized Linear Mixed Models with a Gamma Response . . . .
7.2.1 CRD: Estrus Induction in Pelibuey Ewes . . . . . . . . . . . . .
7.2.2 Randomized Complete Block Design (RCBD):
Itch Relief Drugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.3 Factorial Design: Insect Survival Time . . . . . . . . . . . . . .
7.2.4 A Split Plot with a Factorial Structure on a Large
Plot in a Completely Randomized Design (CRD) . . . . . . .
7.3
Survival Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 CRD: Aedes aegypti . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.3 RCBD: Aedes aegypti . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Linear Mixed Models for Categorical and Ordinal
Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3
Cumulative Logit Models (Proportional Odds Models) . . . . . . . .
8.3.1 Complete Randomize Design (CRD)
with a Multinomial Response: Ordinal . . . . . . . . . . . . . . .
8.3.2 Randomized Complete Block Design (RCBD)
with a Multinomial Response: Ordinal . . . . . . . . . . . . . . .
8.4
Cumulative Probit Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5
Effect of Judges’ Experience on Canned Bean Quality
Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6
Generalized Logit Models: Nominal Response Variables . . . . . . .
8.6.1 CRDs with a Nominal Multinomial Response . . . . . . . . .
8.6.2 CRD: Cheese Tasting . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Linear Mixed Models for Repeated Measurements . . . .
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2
Example of Turf Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3
Effect of Insecticides on Aphid Growth . . . . . . . . . . . . . . . . . . .
9.4
Manufacture of Livestock Feed . . . . . . . . . . . . . . . . . . . . . . . . .
9.5
Characterization of Spatial and Temporal Variations
in Fecal Coliform Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6
Log-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6.1 Emission of Nitrous Oxide (N2O) in Beef Cattle
Manure with Different Percentages of Crude Protein
in the Diet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
279
279
280
280
282
284
287
291
293
295
297
301
306
321
321
322
323
324
328
336
338
348
350
353
357
374
377
377
378
382
387
390
395
397
Contents
Effect of a Chemical Salt on the Percentage Inhibition
of the Fusarium sp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.8
Carbon Dioxide (CO2) Emission as a Function of Soil
Moisture and Microbial Activity . . . . . . . . . . . . . . . . . . . . . . . .
9.9
Effect of Soil Compaction and Soil Moisture on Microbial
Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.10 Joint Model for Binary and Poisson Data . . . . . . . . . . . . . . . . . .
9.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
9.7
398
402
407
409
413
416
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
Chapter 1
Elements of Generalized Linear Mixed
Models
1.1
Introduction to Linear Models
Linear models are commonly used to describe and analyze datasets from different
research areas, such as biological, agricultural, social, and so on. A linear model aims
to best represent/describe the nature of a dataset. A model is usually made up of
factors or a series of factors that can be nominal or discrete variables (sex, year, etc.)
or continuous variables (age, height, etc.), which have an effect on the observed data.
Linear models are the most commonly used statistical models for estimating and
predicting a response based on a set of observations.
Linear models get their name because they are linear in the model parameters.
The general form of a linear model is given by
y = Xβ þ ε
ð1:1Þ
where y is the vector of dimension n × 1 observed responses, X is the design matrix
of n × ( p + 1) fixed constants, β is the vector of ( p + 1) × 1 parameters to be
estimated (unknown), and ε is the vector of n × 1 random errors. Linearity arises
because the mean response of vector y is linear to the vector of unknown parameters
β. Mathematically, this is demonstrated by obtaining the first derivative of the
predictor with respect to β, and, if after derivation it is still a function of any of the
beta parameters, then the model is said to be nonlinear; otherwise, it is a linear
model. In this case, the derivative of the predictor (1.1) with respect to beta is equal
to X, so, mathematically, the model in (1.1) is linear, since after derivation, the
predictor no longer depends on the β parameters.
Several models used in statistics are examples of the general linear model
y = Xβ + e. These include regression models and analysis of variance (ANOVA)
models. Regression models generally refer to those in which the design matrix X is
© The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8_1
1
2
1
Elements of Generalized Linear Mixed Models
of a full column rank, whereas in analysis of variance models, the design matrix X is
not of a full column rank. Some linear models are briefly described in the following
sections.
1.2
Regression Models
Linear models are often used to model the relationship between a variable, known as
the response or dependent variable, y, and one or more predictors, known as
independent or explanatory variables, X1, X2, ⋯, Xp.
1.2.1
Simple Linear Regression
Consider a model in which a response variable y is linearly related to an explanatory
variable X1 via
yi = β0 þ β1 X 1i þ εi
where εi are uncorrelated random errors (i = 1, 2, ⋯, n) which are commonly
assumed to be normally distributed with mean 0 and variance constant σ 2 > 0,
εi ~ N(0, σ 2). If X11, X12, ⋯, X1n are constant (fixed), then this is a general linear
model y = Xβ + e where
yn × 1 =
y1
y2
⋮
yn
, Xn × 2 =
1
1
⋮
1
X 11
X 12
⋮
X 1n
, β2 × 1 =
β0
, en × 1 =
β1
E1
E2
⋮
En
Example Let us consider the relationship between the performance test scores and
tissue concentration of lysergic acid diethylamide commonly known as LSD (from
German Lysergsäure-diethylamid) in a group of volunteers who received the drug
Table 1.1 Average
mathematical test
scores and LSD tissue
concentrations
Tissue concentration of LSD
1.17
2.97
3.26
4.69
5.83
6.00
6.41
Mathematical average
78.93
58.20
67.47
37.47
45.65
32.92
29.97
1.2
Regression Models
3
Table 1.2 Results of the
simple regression analysis
(a) Type III tests of fixed effects
Effect
Num DF
Den DF
Conc
1
5
(b) Parameter estimates
Standard
Effect
Estimate
error
Intercept 89.1239
7.0475
Conc
-9.0095
1.5031
50.7763
32.1137
Scale
σ2
F-value
35.93
DF
5
5
.
t-value
12.65
-5.99
.
Pr > F
0.0019
Pr > |t|
<0.0001
0.0019
.
(Wagner et al. 1968). The average scores on the mathematical test and the LSD
tissue concentrations are shown in Table 1.1.
The components of this regression model are as follows:
Distribution: yi N ηi , σ 2
Linear predictor: ηi = β0 þ β1 × Conci
Link function: μi = ηi ðidentityÞ
The syntax for performing a simple linear regression using the GLIMMIX
procedure in Statistical Analysis Software (SAS) is as follows:
proc glimmix;
model y= X1/solution;
run;
Part of the results is shown in Table 1.2. The analysis of variance (item a)
indicates that drug concentration has a significant effect on average mathematical
performance (P = 0.0019). The estimates of the regression model parameters (item b)
are β0 and β1, and the mean squared error (MSE scale) is shown in Table 1.2(b)
under “Parameter estimates.”
With these results, the linear predictor ðηi Þ that predicts the average mathematical
performance as a function of LSD concentration is as follows:
ηi = 89:124 - 9:01 × Conci
This means that we can predict the average mathematical performance of an
individual for whom we need to know the LSD concentration (Conci) to be applied.
From the estimated parameters, we can say that there is a negative relationship
between LSD concentration and mathematical score. Figure 1.1 clearly shows that
an increase in drug supply has a negative effect on the mathematical score of the
youth. This fitted model explains 87.7% of the variability in the data (Fig. 1.1).
4
1
Elements of Generalized Linear Mixed Models
90
80
Average score
70
y = 89.124 -9.0095*Conc
R² = 0.8778
60
50
40
30
20
10
0
1
2
3
4
5
6
7
Concentration (LSD)
Fig. 1.1 Relationship between applied drug concentration and the mathematical score of the youth
Adjusted model of the relationship between the average score and LSD
concentration.
1.2.2
Multiple Linear Regression
Suppose that a response variable y is linearly related to several independent variables
X1, X2, ⋯, Xp such that
yi = β0 þ β1 X i1 þ β2 X i2 þ ⋯ þ βp X ip þ εi
for i = 1, 2, ⋯, n. Here, εi are uncorrelated random errors (i = 1, 2, ⋯, n) normally
distributed with a zero mean and constant variance σ 2, i.e., εi ~ N(0, σ 2). If the
explanatory variables are fixed constants, then the above model belongs to a general
linear model of the form y = Xβ + ε, as can be seen below:
yn × 1 =
y1
y2
⋮
yn
εn × 1 =
E1
E2
⋮
En
1
1
, X n × ðpþ1Þ =
⋮
1
X 11
X 21
⋮
X n1
X 12 ⋯
X 22 ⋯
⋮
X n2 ⋯
X 1p
X 2p
⋮
X np
, βp × 1 =
β0
β1
β2
⋮
βp
,
1.2
Regression Models
5
Table 1.3 Body weight (kilograms) and its relationship with circumference (centimeters) and heart
length (centimeters) of seven young bulls
Bull
Weight (kilograms)
Circumference (centimeters)
Length (centimeters)
1
480
175
128
2
450
177
122
3
480
178
124
4
500
175
131
5
520
186
131
6
510
183
130
7
500
185
124
A regression analysis can be used to assess the relationship between explanatory
variables and the response variable. It is also a useful tool for predicting future
observations or simply describing the structure of the data.
Example Let us to fit a regression model of the relationship between body weight
and heart girth and length of the hearts of seven young bulls from the data shown in
Table 1.3.
The components of this multiple regression model are as follows:
Distribution: yi N ηi , σ 2
Linear predictor: ηi = β0 þ X i1 β1 þ X i2 β2
Link function: μi = ηi ðidentityÞ
The syntax for performing a multiple regression using the GLIMMIX procedure
in SAS, assuming that there is no interaction between bull heart girth (X1) and length
(X2), is shown below:
proc glimmix;
model y = X1 X2/solution cl;
run;
Based on the regression model specifications, the option “solution cl” prompts
GLIMMIX to provide the value of the estimated parameters and their respective
confidence intervals. Other useful options available are “htype = 1, 2, and 3,” which
refer to the sum of squares of types I, II, and III. The type III fixed effects tests in
(a) of Table 1.4 indicate that there is a linear relationship between heart length (size)
and weight in young bulls. The estimated parameters with their respective confidence intervals β0 , β1 , β2 as well as the MSE (scale) of the fitted regression model
are listed below in (b).
Note that in a linear model, the parameters are linearly entered, but the variables
do not necessarily have to be linear. For example, consider the following two
examples:
yi = β0 þ β1 X i1 þ β2 logðX i2 Þ þ ⋯ þ βk X ik þ Ei
6
1
Elements of Generalized Linear Mixed Models
Table 1.4 Results of the multiple regression analysis
(a) Type III tests of fixed effects
Effect
Num DF
X1
1
X2
1
(b) Parameter estimates
Effect
Estimate Standard error
Intercept -495.01 225.87
X1
2.2573
1.0739
X2
4.5808
1.4855
Scale
139.51
98.6518
Den DF
DF
.
t-value
-2.19
2.10
3.08
.
Pr > F
0.1034
0.0368
F-value
4.42
9.51
Pr > |t|
0.0935
0.1034
0.0368
.
α
0.05
0.05
0.05
.
Lower
-1122.13
-0.7243
0.4564
.
Upper
132.10
5.2388
8.7053
.
β
yi = β0 þ X i11 þ β2 X i2 þ ⋯ þ βk expðX ik Þ þ Ei
The first example is a linear model, whereas the second one is not, since its
β
derivatives do not depend on the beta coefficients, with the exception of the term X i11
β1
whose derivative is equal to X i1 logðX i1 Þ . This clearly shows that the second
example is a nonlinear model because the derivative of the predictor depends on β1.
1.3
1.3.1
Analysis of Variance Models
One-Way Analysis of Variance
Consider an experiment in which you want to test t treatments (t > 2), to the level of
the ith treatment with ni experimental units that are selected and randomly assigned
to the ith treatment. The model describing this experiment is as follows:
yij = μ þ τi þ Eij
for i = 1, 2, ⋯, t and j = 1, 2, ⋯, ni . Here, Eij are the uncorrelated random errors with
normal distribution with a zero mean and a variance constant σ 2 (εij ~ N(0, σ 2)). If the
treatment effects are considered as fixed constants (drawn from a finite number), then
this model is a special case of the general linear model (1), with the total number of
experimental units n =
t
ni .
i
1.3
Analysis of Variance Models
7
Table 1.5 Biomass
production of the
three types of bacteria
Bacteria A
12
15
9
Table 1.6 Analysis of
variance
Sources of variation
Bacteria type
Error
Total
Bacteria B
20
19
23
Bacteria C
40
35
42
Degrees of freedom
t-1=3-1=2
t(r - 1) = 3 × 2 = 6
tr - 1 = 3 × 3 - 1 = 8
In matrix terms, the information under this design of experiment is equal to:
yn × 1 =
y11
y12
⋮
ytnt
en × 1 =
ε11
ε12
E13
⋮
Etnt
, Xn × ðtþ1Þ =
1n 1
1n 2
⋮
1n t
1n 1
0n 2
⋮
0n t
0n1
1n2
⋮
0n t
⋯
⋯
⋱
⋯
0n1
0n2
⋮
1nt
, βtþ1 × 1 =
μ
τ1
τ2
⋮
τt
,
where 1ni is the vector of ones of order ni and 0ni is the vector of zeros of order ni.
Note that the matrix Xn×(t+1) is not of a full column rank because its first column can
be obtained as a linear combination of its remaining columns.
Example Assume that measurements of the biomass produced by three different
types of bacteria are collected in three separate Petri dishes (replicates) in a glucose
broth culture medium for each bacterium (Table 1.5).
The sources of variation and degrees of freedom (DFs) for this experiment are
shown in Table 1.6.
The components for this one-way model, assuming that each of the response
variable yij is normally distributed, are as follows:
Distribution: yij N μij , σ 2
Linear predictor: ηi = α þ τi
Link function: μi = ηi ðidentityÞ
where yij is the response observed at the jth repetition in the ith bacterium, ηi is the
linear predictor, α is the intercept (the grand mean), and τi is the fixed effect due to
the type of bacterium.
8
1
Table 1.7 Results of the
one-way analysis of variance
Elements of Generalized Linear Mixed Models
(a) Fit statistics
-2 Res log likelihood
33.36
AIC (Akaike information criterion) (smaller is better)
41.36
AICC (Corrected Akaike information criterion) (smaller 81.36
is better)
BIC (Bayesian information criterion) (smaller is better) 40.52
CAIC (Consistent Akaike's information criterion)
44.52
(smaller is better)
HQIC (Hannan and Quinn information criterion)
38.02
(smaller is better)
Pearson’s chi-square
52.67
Pearson’s chi-square / DF
8.78
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Bacteria
64.95
<0.0001
The SAS syntax for a one-way analysis of variance (ANOVA) is as follows:
proc glimmix data=biomass;
class bacteria;
model y = bacteria;
lsmeans bacteria/lines;
run;
Similar to “proc glm” or “proc mixed,” the “class” command allows to define the
type of class variables (categorical or nominal) to be included in the model; in this
case, for the class variable “bacteria,” the “model” command allows to declare (list)
the response variable “y” and all the class or continuous variables that enter the
model, whereas the “lsmeans” command asks GLIMMIX to estimate the means of
the treatments and the “lines” option allows to make a comparison of means. Part of
the results is presented below.
By default, “proc GLIMMIX” provides the fit statistics (information criteria),
which are extremely useful for comparing or choosing a model that explains the
largest possible proportion of variation present in a dataset, i.e., the best-fit model
(part (a) of Table 1.7). The statistic “-2 res log likelihood” is most useful when
comparing nested models, and the rest of the statistics is useful for comparing
models that are not necessarily nested. The mean squared error (MSE) in GLIMMIX
is given as the statistic “Pearson′s chi - square/DF.” In this analysis, this value is
8.78. σ 2 = MSE = 8:78 . In part (b), the analysis of variance indicates that at least
one type of bacterium produces a different biomass (P < 0.0001). That is, the null
hypothesis is rejected (H0 : τA = τB = τC) at a significance level of 5%.
The estimated least squares (LS) means obtained with “lsmeans” are tabulated
under the “Estimate” column with their standard errors in the “Standard error”
column of Table 1.8. These estimated means were obtained (by default) with
Fisher’s LSD (least significant difference).
1.3
Analysis of Variance Models
9
Table 1.8 Means and estimated standard errors of the one-way model
Least squares means of bacteria
Bacteria
Estimate
A
12.0000
B
20.6667
C
39.0000
Table 1.9 Comparison of the
means (LSD) in the one-way
model
Standard error
1.7105
1.7105
1.7105
DF
t-value
7.02
12.08
22.80
Pr > |t|
0.0004
<0.0001
<0.0001
T grouping of the least squares means of bacteria (α = 0.05)
LS means with the same letter are not significantly different
Bacteria
Estimate
C
39.0000
A
B
20.6667
B
A
12.0000
C
Finally, Table 1.9 presents a comparison of the means obtained with “lines” and
indicates that bacteria type C has a better fermentative conversion of glucose to lactic
acid compared to bacteria types B and A. Equal letters per column indicate that they
are statistically equal.
1.3.2
Two-Way Nested Analysis of Variance
Let us consider an experiment with two factors, A and B, in which each level of B is
nested within a level of factor A, that is, each level of factor B appears within a level
of factor A. Then, the model that describes this experiment is as follows:
yijk = μ þ αi þ βjðiÞ þ Eijk
for i = 1, 2, ⋯, a; j = 1, 2, ⋯, bi; and k = 1, 2, ⋯, nij. In this model, μ is the overall
mean, αi represents the effect due to the ith level of factor A, and βj(i) represents the
effect of the jth level of factor B nested within the ith level of factor A. Assuming
that all factors are fixed, and that the errors εijk are normally distributed, that is
εijk~N(0, σ 2), this model is the general linear model of the form y = Xβ + e. For
example, suppose that you have a = 3 levels of factor A, b = 2 levels of factor B, and
nij = 2, then the vectors and matrices have the following form:
10
1
y=
y111
y112
y121
y122
y211
y212
y221
y222
y311
y312
y321
y322
β=
μ
α1
α2
α3
β11
β12
β21
β22
β31
β32
, X=
,
e=
1 1
1 1
1 1
1 1
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
E111
E112
E121
E122
E211
E212
E221
E222
E311
E312
E321
E322
0
0
0
0
1
1
1
1
0
0
0
0
Elements of Generalized Linear Mixed Models
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
,
:
Example Suppose that a researcher was studying the assimilation of fluorescently
labeled proteins in rat kidneys and wanted to know whether his two technicians,
technician A and technician B, were performing the procedure consistently. Technician A randomly chose three rats, and technician B randomly chose three other
rats, and each technician measured the protein assimilation in each rat. Since rats are
expensive and measurements are cheap, both technicians measured protein assimilation at various random locations in the kidneys of each rat (Table 1.10).
When performing a nested ANOVA, we are often interested in testing the null
hypothesis (Ho : τA = τB ). As in this example, we do not wish to test whether the
subgroups (rats within technicians) are significantly different, since the goal is to
prove that both technicians are performing their jobs adequately. The sources of
variation and degrees of freedom are shown in Table 1.11.
The components of this two-way model, assuming that the response variable yij is
normally distributed, are as follows:
1.3
Analysis of Variance Models
Table 1.10 Levels of protein
assimilation in the rat kidneys
measured by both technicians
Technician A
Rat1
Rat2
1.119
1.045
1.2996
1.1418
1.5407
1.2569
1.5084
0.6191
1.6181
1.4823
1.5962
0.8991
1.2617
0.8365
1.2288
1.2898
1.3471
1.1821
1.0206
0.9177
Table 1.11 Sources of
variation and degrees
of freedom of the two-way
nested design
Sources of variation
Technician
Rat (technical)
Error
Total
11
Technician B
Rat4
Rat5
1.3883
1.3952
1.104
0.9714
1.1581
1.3972
1.319
1.5369
1.1803
1.3727
0.8738
1.2909
1.387
1.1874
1.301
1.1374
1.3925
1.0647
1.0832
0.9486
Rat3
0.9873
0.9873
0.8714
0.9452
1.1186
1.2909
1.1502
1.1635
1.151
0.9367
Rat6
1.2574
1.0295
1.1941
1.0759
1.3249
0.9494
1.1041
1.1575
1.294
1.4543
Degrees of freedom
a-1=2-1=1
a(b - 1) = 2(3 - 1) = 4
ab(r - 1) = 2 × 3(10 - 1) = 54
abr - 1 = 2 × 3 × 10 - 1 = 59
Distribution: yij N μij , σ 2
Linear predictor: ηij = α þ τi þ βðτÞjðiÞ
Link function: μi = ηi ðidentityÞ
where yij is the level of assimilation of the fluorescent protein obtained from rat j by
technician i, α is the intercept, τi is the fixed effect due to the technician, and β(τ)j(i) is
the nested effect of rat j within technician i.
The SAS commands for the main effects of factor A and factor B nested within A
are as follows:
proc glimmix data=rata nobound;
class technician rat rep;
model protein=technical rat(technical);
lsmeans technician rat(technician)/lines;
run;
Part of the results is shown in Table 1.12. The results indicate that there is minimum
variability of the technicians since the value of the mean squared error
(Pearson′s chi - square/DF) is 0.04 (part (a)). This means that the variance between
group means is smaller than would be expected. The analysis of variance in part
(b) indicates that there is no difference in the measurement of fluorescent proteins in
the rats between technicians (P = 0.3065). Since there is variation between rats in the
12
Table 1.12 Fit statistics of
the two-way nested design
1
Elements of Generalized Linear Mixed Models
(a) Fit statistics
-2 Res log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson’s chi-square
Pearson’s chi-square / DF
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
Technician
1
Rat (technical)
-12.39
1.61
4.04
15.53
22.53
6.98
1.95
0.04
F-value
1.07
3.98
Pr > F
0.3065
0.0067
Table 1.13 Comparison of the means (LSD) in the nested model
(a) Technical least squares means
Technician
Estimate
Standard error
DF
A
1.2110
0.03466
B
1.1604
0.03466
(b) T grouping of the technical least squares means (α = 0.05)
LS means with the same letter are not significantly different
Technician
Estimate
A
1.2110
B
1.1604
Pr > |t|
<0.0001
<0.0001
t-value
34.94
33.48
A
A
A
average protein uptake, it is to be expected that between rats within technicians, there
are mean differences in the protein uptake (P = 0.0067).
In Table 1.13 part (a), the values of the least squares means tabulated under the
“Estimate” column are shown with their respective “Standard errors.” It can be seen
that rats under technician A have statistically the same mean protein uptake as do rats
under technician B (part (b)).
Comparison of means for rat subgroups under both technicians showed similar
means for rats under technician A but different means for rats under technician B
(part (a) and (b), Table 1.14).
1.3
Analysis of Variance Models
13
Table 1.14 Comparison of the means (LSD) of the subgroups nested within technicians
(a) Least squares means of rats (technical)
Technician
Rat
Estimate
Standard error
DF
A
5
1.2187
0.06003
A
1.2302
0.06003
A
1.1841
0.06003
B
1
1.3540
0.06003
B
1.0670
0.06003
B
1.0602
0.06003
(b) T grouping of the least squares means (α = 0.05) of rats (technical)
LS means with the same letter are not significantly different
Technician
Rat
Estimate
B
1
1.3540
A
1.2302
B
A
5
1.2187
B
A
1.1841
B
B
1.0670
B
B
1.0602
B
1.3.3
t-value
20.30
20.49
19.72
22.56
17.77
17.66
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
A
A
A
A
Two-Way Analysis of Variance with Interaction
This experiment is used when one wishes to test two factors A and B, with a levels of
factor A and b levels of factor B. In this experiment, both factors are crossed, this
means that each level of A occurs in combination with each level of factor B. The
model with interaction is given by:
yijk = μ þ αi þ βij þ γ ij þ Eijk
for i = 1, 2, ⋯, a; j = 1, 2, ⋯, b; k = 1, 2, ⋯, nij; and εijk ~ N(0, σ 2). If all the
parameters of the model are fixed, then this model can be expressed as y = Xβ + e.
For this model with a = 3, b = 2, and nij = 3, the matrix expression has the form:
14
1
y=
y111
y112
y113
y121
y122
y123
y211
y212
y213
y221
y222
y223
y311
y312
y313
y321
y322
y323
β=
μ
α1
α2
α3
β1
β2
γ 11
γ 12
γ 21
γ 22
γ 31
γ 32
, X=
,
e=
1 1
1 1
1 1
1 1
1 1
1 1
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
E111
E112
E113
E121
E122
E123
E211
E212
E213
E221
E222
E223
E311
E312
E313
E321
E322
E323
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
Elements of Generalized Linear Mixed Models
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
,
Example This experiment consisted of developing an in vitro efficacy test for selftanning formulations. Two brands, 1 = erythrulose, 2 = dihydroxyacetone (factor
A), and three formulations, 1 = solution, 2 = gel, and 3 = cream (factor B), were
tested with four replicates for each condition according to Jermann et al. (2001).
Total color change was measured for each of the combination conditions. The
dataset is shown in Table 1.15.
1.3
Analysis of Variance Models
15
Table 1.15 Color change (Y ) in each of the brands and formulations
Brand
1
1
1
1
1
1
1
1
1
1
1
1
Formulation
1
1
1
1
Y
16.79
12.68
12.47
11.67
10.23
10.29
8.97
8.51
9.43
9.45
8.86
8.66
Table 1.16 Analysis of variance of the two-way model
with interaction
Brand
Sources of variation
Brand
Formulation
Brand × formulation
Error
Total
Formulation
1
1
1
1
Y
32.85
38.08
30.25
28.41
25.06
21.66
19.86
18.62
25.89
22.96
24.55
24.59
Degrees of freedom
a-1=2-1=1
a-1=3-1=2
(a - 1)(b - 1) = 1 × 2 = 2
ab(r - 1) = 2 × 3 × 3 = 18
abr - 1 = 2 × 3 × 4 - 1 = 23
For this two-way model, assuming that the response variable yijk has a normal
distribution, the components are as follows:
Distribution: yijk N μijk , σ 2
Linear predictor: ηij = μ þ αi þ βj þ γ ij
Link function: μij = ηij ðidentityÞ
where yijk is the color change observed at the kth repetition at the ith level of factor A
and at the jth level of factor B, μ is the intercept (the overall mean), αi is the fixed
effect due to the level of factor A (mark), βj represents the fixed effect of the level of
factor B (type of formulation), and γ ij is the fixed effect due to the interaction
between the brand and formulation. Table 1.16 shows the sources of variation and
degrees of freedom.
The following code in GLIMMIX in SAS allows us to estimate the main effects
and the interaction:
proc glimmix;
class brand formulation;
model y = brand|formulacion;
lsmeans brand|formulacion/lines;
run;
16
1
Elements of Generalized Linear Mixed Models
Table 1.17 Results of the analysis of variance of the two-way model with interaction
(a) Fit statistics
-2 Res log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson’s chi-square
Pearson’s chi-square / DF
(b) Type III tests of fixed effects
Effect
Num DF
Brand
1
Formulation
Brand × formulation
90.20
104.20
115.40
110.43
117.43
105.06
99.61
5.53
Den DF
Pr > F
<0.0001
<0.0001
0.0231
F-value
257.04
22.99
4.68
Table 1.18 Means and standard errors of the tanning brand
Least squares means of the brand
Brand
Estimate
Standard error
DF
1
10.6675
0.6791
26.0650
0.6791
T grouping of the least squares means (α =0.05)
LS means with the same letter are not significantly different
Brand
Estimate
26.0650
1
10.6675
Pr > |t|
<0.0001
<0.0001
t-value
15.71
38.38
A
B
Part of the results is shown below. Of all the fit statistics in (a) of Table 1.17, the
value that we are interested in highlighting in this analysis is “Pearson′s chi square/DF,” which corresponds to the mean squared error (MSE), even though we
are evaluating different possible models for this given dataset. The value of the MSE
is 5.53. The type III fixed effects tests, in part (b) of Table 1.17, indicate that the type
of brand (P < 0.0001), formulation (P < 0.0001), and the interaction between both
factors (P = 0.0231) all have a significant effect on the change of self-tanning color.
The least mean squares obtained with “lsmeans” are shown in the Table 1.18 for
the levels of tanning brand factor in Table 1.19 for the levels of tanning brand
formulation and in Table 1.20 for the interaction of both factors. The “lines” option
allows us to make a comparison of means using the LSD method.
The least squares means for the tanning brand factor are given in Table 1.18.
The least squares means for the type of tanning brand formulation are given in
Table 1.19.
1.3
Analysis of Variance Models
17
Table 1.19 Means and standard errors of the tanning brand formulation
Least squares means for the tanning brand formulation
Formulation
Estimate
Standard error
DF
t-value
1
22.9000
0.8317
27.53
15.4000
0.8317
18.52
16.7988
0.8317
20.20
T grouping of the least squares means (α = 0.05) of the tanning brand formulation
LS means with the same letter are not significantly different
Formulation
Estimate
1
22.9000
A
16.7988
B
B
15.4000
B
Pr > |t|
<0.0001
<0.0001
<0.0001
Table 1.20 Comparison of the means of the interaction of both factors
T grouping of the least squares means (α =0.05) of the marca*formulation
LS means with the same letter are not significantly different
Brand
Formulation
Estimate
1
32.3975
24.4975
21.3000
1
1
13.4025
1
9.5000
1
9.1000
A
B
B
C
D
D
The hypothesis test for the interaction should be tested first, and only if the
interaction effect is not significant, should the main effects be tested. If the interaction is significant, then tests for the main effects are meaningless. The interaction
analysis shows that brand 2 (dihydroxyacetone), in all three formulations, shows a
greater change compared to brand 1 (erythrulose).
Now, considering the previous model without interaction (γ 11 = γ 12 = ⋯ = γ 32 = 0)
where factor A has a levels and factor B has b levels, the model without interaction is
given by:
yijk = μ þ αi þ βj þ Eijk
for i = 1, 2, ⋯, a; j = 1, 2, ⋯, b; k = 1, 2, ⋯, nij; and εijk ~ N(0, σ 2). The model
without interaction with a = 3, b = 2, and nij = 3 reduces to:
18
y=
1
y111
y112
y113
y121
y122
y123
y211
y212
y213
y221
y222
y223
y311
y312
y313
y321
y322
y323
,
X=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
Elements of Generalized Linear Mixed Models
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
, β=
μ
α1
α2
α3
β1
β2
, e=
E111
E112
⋮
E322
E323
Note that the design matrix for the model without interaction is the same as that
for the model with interaction, except that the last six columns are removed.
Let us assume that the interaction effect is not significant. The following SAS
code estimates the main effects of both factors. Running the program and analysis is
left as practice for the readers.
proc glimmix;
class brand formulation;
model y = formula brand;
lsmeans brand formulation/lines;
run;
1.4
Analysis of Covariance (ANCOVA)
Consider an experiment to compare t ≥ 2 treatments after adjusting for the effects of
a covariate x. The model for an analysis of covariance is given by:
yij = μ þ τi þ βi xij þ Eij
for i = 1, 2, ⋯, t and j = 1, 2, ⋯, ni Here, Eij are the independent normally distributed
random errors with a zero mean and a variance constant σ 2 > 0. In this model, μ is
the overall mean, τi is the fixed effect of the ith treatment (ignoring the covariates
x’s), βi denotes the slope of the line that relates the response variable y to x for the
ith treatment, and xij are fixed covariates. Assuming t = 3, n1 = n2 = n3 = 3,
we have:
1.4
Analysis of Covariance (ANCOVA)
y=
e=
y11
y12
y13
y21
y22
y23
y31
y32
y33
E11
E12
E13
E21
E22
E23
E31
E32
E33
, X=
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
19
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
x11
x12
x13
0
0
0
0
0
0
0
0
0
0
0
0
x21 0
x22 0
x23 0
0 x31
0 x32
0 x33
, β=
μ
α1
α2
α3
β1
β2
β3
,
The analysis of covariance (ANCOVA), as can be seen, obeys a general linear
model of the form y = Xβ + e .
For example consider a hypothetical study of flower production in two subspecies
of plants. The number of flowers per plant may vary between the subspecies, but,
within each subspecies, flower production may also vary with the size of each plant,
and this relationship may be positive or negative. A positive relationship might arise
if plants with more resources (sunlight, water, nutrients) could invest more energy in
both growth and flower production. A negative relationship could arise if there was a
trade-off between the energy invested in growth and the energy invested in flower
production. In this study, subspecies is a categorical variable and plant size is a
continuous variable (the covariate). Measuring plant size and flower production in
the two subspecies allows the investigation of three different questions:
Is flower production influenced by subspecies?
Is flower production influenced by plant size?
Is the effect of flower production on plant size influenced by subspecies?
Example 1. The central question in plant reproductive ecology is how hermaphroditic plant species allocate resources to male and female structures. A study
conducted to address this question counted the number of stamens (male structures
that produce pollen) and ovules (female structures that when fertilized by a pollen
grain will become seeds) in the flowers of “prairie larkspur” plants in two
populations in southeastern Minnesota. The total number of flowers produced by
each plant was also determined to assess whether plant size affected ovule production per flower. The dataset for this example can be found in the Appendix (Data:
Larkspur plants).
20
1
Elements of Generalized Linear Mixed Models
An ANCOVA is appropriate for this study to test the following three null
hypotheses from these data:
(a) There is no difference in the average number of ovules per flower between the
two populations (the main effect).
(b) There is no effect of plant size on the average number of ovules per flower (the
covariate effect).
(c) The effect of plant size on the mean number of ovules per flower did not differ
between the study sites (the interaction effect).
The components of the ANCOVA model, assuming that the response variable yijk
is normally distributed, are as follows:
Distribution: yij N μijk , σ 2
Linear predictor: ηij = μ þ τi þ plantaðτÞjðiÞ þ βi X ij - X ::
Link function : μij = ηij ðidentityÞ
where yij is the number of ovules observed in the jth plant of the ith population, μ is
the overall mean,τi is the fixed effect due to the population i, planta(τ)j(i) is the
random effect due to the plant j in the population i, βi is the slope of the population i,
X :: is the overall mean of the size of all plants, and Xij is the plant size i in the
population j. The ANCOVA results (sources of variation and degrees of freedom)
are shown in Table 1.21.
The basic syntax in GLIMMIX for analysis of covariance with different slopes is
as follows:
proc glimmix;
class poblacion plant;
model ovules = population xbar population*xbar/ddfm=satterthwaite;
random plant(population);
lsmeans population/lines;
run;
Table 1.21 Analysis of covariance
Sources of variation
Population
β (depending on the size of the plant)
Population β
Error
Total
Degrees of freedom
t-1=2-1=1
1
1
t
i=1
t
i=1
r i - 1 - t - 1 = 75
r i - 1 = 79 - 1 = 78
1.4
Analysis of Covariance (ANCOVA)
Table 1.22 Results of the
analysis of covariance for the
two populations of larkspur
plants
21
(a) Covariance parameter estimates
Cov Parm
Estimate
Standard error
Plant (population)
12.7951
2.2416
Residual
0.9321
.
(b) Type III tests of fixed effects
Effect
Num DF Den DF F-value Pr > F
Population
1
7.32
0.0084
Center
1
16.39
0.0001
Center x population 1
7.81
0.0066
(c) Least squares means of population
Standard
tPopulation Estimate error
DF value Pr > |t|
Cedar.cr
20.3538 0.6062
33.58 <0.0001
St. Croix
22.7596 0.6502
35.00 <0.0001
(d) T grouping of the least squares means (α = 0.05) of
population
LS means with the same letter are not significantly different
Population
Estimate
St. Croix
22.7596
A
Cedar.cr
20.3538
B
In the above syntax, the “class” command lists all classes or categorical variables,
except the covariate (continuous variable), which – in this case – is a variable
centered by the average of the size of all plants xbar = X ij - X :: : The options
“ddfm” and “lines” invoke proc GLIMMIX to do a degree-of-freedom correction
using the Satterthwaite method and a comparison of the means using the LSD
method. Part of the results is shown in Table 1.22.
The estimates of the variance components (part (a)) due to plant and withintreatment variability are σ 2plantaðpoblacionÞ = 12:795 and σ 2 = MSE = 0:9321, respectively. The analysis of variance in (b) showed that there is a significant effect
between the two populations (P = 0.0084), plant size (P = 0.0001) and plant size
is influenced by subspecies (interaction) on the average number of ovules
(P = 0.0066) per flower. The estimated means and their respective standard errors
of the average number of ovules for both populations are tabulated in the “Estimate”
column in part (c), as well as the comparison of the means in part (d).
If in the previous model we assume that the slopes were equal (β1 = β2), then the
ANCOVA reduces to:
yij = μ þ τi þ β X ij - X :: þ Eij
The ANCOVA model, with t = 3, n1 = n2 = n3 = 3 for this case (equality of
slopes) reduces to:
22
1
y=
y11
y12
y13
y21
y22
y23
y31
y32
y33
, X=
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
Elements of Generalized Linear Mixed Models
x11
x12
x13
x21
x22
x23
x31
x32
x33
, β=
μ
α1
α2
α3
β
, ε=
E11
E12
E13
E21
E22
E23
E31
E32
E33
:
The basic syntax using GLIMMIX for an analysis of covariance with equal slopes
is as follows:
proc glimmix;
class poblacion plant;
model ovules = population xbar/ddfm=satterthwaite;
random plant(population);
lsmeans population/lines;
run;
So far, we have exemplified the general linear model of the form y = Xβ +e. In the
following, some characteristics of a linear mixed model (LMM) will be described.
1.5
1.5.1
Mixed Models
Introduction
Linear mixed models (LMMs) are appropriate for analyzing continuous response
variables in which the residuals are normally distributed. These types of models are
well suited for studies of grouped datasets such as (1) students in classrooms,
animals in herds, people grouped by municipality or geographic region, or randomized block experimental designs such as batches of raw materials for an industrial
process and (2) longitudinal or repeated measures studies, in which subjects are
measured repeatedly over time or under different conditions. These designs occur
in a wide variety of settings: biology, agriculture, industry, and socioeconomic
sciences. LMMs provide researchers with powerful and flexible analytical
tools for these types of data.
The name linear mixed models comes from the fact that these models are linear in
the parameters and that the covariates, or independent variables, may involve a
combination of fixed and random effects. “Fixed effects” can be associated with
continuous covariates, such as weight in kilograms of an animal, maize yield in tons
per hectare, and reference test score or socioeconomic status, which will carry a
continuous range of values, or with factors, such as gender, variety, or group
1.5
Mixed Models
23
treatment, which are categorical. Fixed effects are unknown constant parameters
associated with continuous covariates or levels of the categorical factors in an LMM.
The estimation of these parameters in LMMs is generally of intrinsic interest because
they indicate the relationship of the covariates with the continuous response variable.
When the levels of a factor are drawn from a large enough sample such that each
particular level is not of interest (e.g., classrooms, regions, herds, or clinics that are
randomly sampled from a population), the effects associated with the levels of those
factors can be modeled as random effects in an LMM. “Random effects” are
represented by random (unobserved) variables that we generally assume to have a
particular distribution, with normal distribution being the most common.
Mixed models are extremely useful because they allow us to work on (address)
two important aspects:
1. From a statistical point of view, biological data are often structured in a way that
does not satisfy the assumption of independence of the dataset. Examples include
the following:
(a) Multiple measurements of the same subject/organism
(b) Experiments organized into spatial blocks
(c) Observational data in which multiple investigations were conducted in different locations
(d) Synthesis of data from similar experiments that were performed by different
researchers
2. From a biological perspective, the processes being measured can be affected by
multiple sources of variation, often occurring at different spatial or temporal
scales. We are interested in using statistical methods that can model multiple
sources of stochasticity, at multiple scales, so that we can measure the relative
magnitude of the different sources of variation and determine which predictors
explain variation at different scales.
1.5.2
Mixed Models
The matrix notation for a mixed model is highly similar to that for a fixed effects
(systematic) model. The main difference is that, instead of using only one design
matrix to explain the entire model in its systematic part, the matrix notation for a
mixed model uses at least two design matrices: a design matrix X to describe the
fixed effects in the model and a design matrix Z to describe the random effects in the
model. The fixed effects design matrix X is constructed in the same way as a general
linear fixed effects model (y = Xβ + ε ). X has a dimension of n × ( p + 1), where n is
the number of observations in the dataset and p + 1 is the number of parameters of
fixed effects in the model to be estimated. The design matrix for the random effects
Z is constructed in the same way as the construction of the design matrix for
fixed effects, but now for the random effects. The Z matrix has a dimension of
n × q, where q is the number of coefficients of random effects in the model.
24
1
Elements of Generalized Linear Mixed Models
In matrix notation, a linear mixed model can be represented as
y=
Sistematic
random
Experimental Error
Xβ
þ Zb þ
ε
ð1:2Þ
b N ð0, GÞ and ε N ð0, RÞ
where y is the vector of n × 1 observations, β is the vector of ( p + 1) × 1 fixed effects,
b is the vector of random effects of q × 1, ε is the vector of n × 1 random error terms,
X is the design matrix of n × ( p + 1) for fixed effects related to observations at β, and
Z is the design matrix n × q for the random effects (b) related to observations at b.
Assuming that both b and ε are uncorrelated random variables with a zero mean
and variance–covariance matrices G and R, respectively, we have
E ðbÞ = 0, E ðε Þ = 0
VarðbÞ = G, Varðε Þ = R
Covðb, εÞ = 0
It is not difficult to verify that Var(y) = Var(Xβ + Zb + ε) is
VarðyÞ = ZGZ0 þ R = V
Matrix V is an important component when working with linear mixed models
(LMMs) because it contains random sources of variation and also defines how such
models differ from ordinary least squares estimation. If the model contains only
random effects, such as a randomized complete block design (RCBD), then matrix G
is the first point of attention. On the other hand, for repeated measures or for spatial
analysis, matrix R is extremely important. Assuming that the random effects (blocks)
have a normal distribution,
b N ð0, GÞ and Varðε Þ = R
Then, the vector of observations y will have a normal distribution, that is,
y~N(Xβ, V). The same model can be written in the probability distribution form in
two different but equivalent ways. The first is the marginal model
y N ðE ½y = Xβ, V = ZGZ0 þ RÞ
ð1:3Þ
In this marginal model, the mean is based only on fixed effects and the parameters
describing the random effects appear (are contained) in the variance and covariance
matrix V (Littell et al. 2006). In general, a structure is imposed in b in terms of
Var(b) = G, and, therefore, marginally, the components of y depend on the structure
in V = ZGZ′ + R.
1.5
Mixed Models
25
The second model is the conditional model
y j b N ðXβ þ Zb, RÞ
ð1:4Þ
In this conditional model, b is distributed as shown in Eq. (1.2) for this parameter.
For LMMs, the two models are exactly the same; but if the response variable is
modeled under a non-normal distribution, then the models are different (Stroup,
2012) and generalized linear mixed models are required.
The fixed effects estimator (β) is useful to obtain the best linear unbiased
estimators (commonly known as BLUEs), whereas the estimator b is useful for
computing the best linear unbiased predictors (commonly known as BLUPs) for the
random effects b. The estimation of the expected value of the marginal LMM (1.3)
allows the estimation of the BLUEs and that of the conditional LMM (1.4), the
BLUPs. The estimators for the BLUEs of β and the BLUPs of b are as follows:
β = XT V - 1 X
-1
XT V - 1 y
b = GZT V - 1 y - β
This solution is efficient when working with small datasets because, in the context
of big data, it is computationally highly demanding since the inverse of matrix V has
to be estimated. For this reason, it is normally used to obtain the solution of the
BLUEs of β and the BLUPs of b, also known as Henderson’s mixed model
equations, which are presented later in this chapter.
1.5.3
Distribution of the Response Variable Conditional
on Random Effects (y| b)
The distribution selected by the researcher from the population under study should
be true or a good approximation that represents the likely distribution of the response
variable. A good representation of the population distribution of a response variable
should not only take into account the nature of the response variable (e.g., continuous, discrete, etc.) and the shape of the distribution but should also provide a good
model for the relationship between the mean and variance. For the distribution of the
dataset, in this chapter, we assume that it is normally distributed with a mean μ and a
variance σ 2 {yij ~ (μ, σ 2)} and, for the random effects, it will assume a normal
distribution with mean 0 and constant variance σ 2b bj 0, σ 2b .
26
1.5.4
1
Elements of Generalized Linear Mixed Models
Types of Factors and Their Related Effects on LMMs
In an LMM, there are two types of factors, namely, fixed factors that make up the
systematic part and random factors that are the stochastic part, and their related
effects on the dependent variable (response). In the following sections, we provide a
brief description of these factors and their implications in the context of an LMM.
1.5.4.1
Fixed Factors
A fixed factor is commonly used in standard analysis of variance (ANOVA) or
analysis of covariance (ANCOVA) models. It is defined as a categorical or classification variable, for which the researcher has included all levels (or conditions) in
the model that are of interest in the study. Fixed factors may include qualitative
covariates, such as gender; classification variables implied by a sampling design,
such as a region or a stratum, or by a study design, such as the method of treatment in
a randomized clinical trial; and so on. The levels of a fixed factor are chosen to
represent specific conditions so that they can be used to define contrasts (or sets of
contrasts) of interest in the research study.
1.5.4.2
Random Factors
A random factor is a classification variable with levels that can be randomly sampled
from a population with different levels of study. All possible levels of a random
factor are not present in the dataset, but this is the intention of the researcher, i.e., to
make inference about the entire population of levels from the selected sample of
these factor levels. Random factors are considered in an analysis such that the
change in the dependent variable across random factor levels can be evaluated and
the results of the data analysis can be generalized to all random factor levels in the
population.
1.5.4.3
Fixed Versus Random Factors
In contrast to fixed factor levels, random factor levels do not represent conditions
specifically chosen to meet the objectives of the study. However, depending on the
objectives of the study, the same factor may be considered as either a fixed factor or a
random factor.
Fixed effects, commonly referred to as regression coefficients or fixed effect
parameters, describe the relationships between the dependent variable and predictor
variables (i.e., fixed factors or continuous covariates) for an entire population of
units of analysis or for a relatively small number of subpopulations defined by the
levels of a fixed factor. Fixed effects may describe the contrasts or differences
1.5
Mixed Models
27
between levels of a fixed factor (e.g., sex between males and females) in the mean
responses for a continuous dependent variable or may describe the effect of a
continuous covariate on the dependent variable. Fixed effects are assumed to be
unknown fixed quantities in an LMM and are estimated based on analysis of the data
collected in a study.
Random effects are random values associated with the levels of a random factor
(or factors) in an LMM. These values, which are specific to a given level of a random
factor, generally represent random deviations from the relationships described by
fixed effects. For example, random effects associated with levels of a random factor
may enter an LMM as random intercepts (random deviations for a given subject or
group as an overall intercept) or as random coefficients (random deviations for a
given subject or group from the total fixed effects) in the model. In contrast to fixed
effects, random effects are represented as stochastic variables in an LMM.
1.5.5
Nested Versus Crossed Factors and Their
Corresponding Effects
When a given level of one factor (random or fixed) can be measured only at a single
level of another factor and not across multiple levels, then the levels of the first factor
are said to be nested within the levels of the second factor. The effects of the nested
factor on the response variable are known as nested effects. For example, suppose
that you want to conduct a particular study at the primary level in a school zone, you
would select schools and classrooms at random. Classroom levels (one of the
random factors) are nested within school levels (another random factor), since
each classroom can appear within a single school.
When a given level of one factor (random or fixed) can be measured across
multiple levels of another factor, one factor is said to be crossed with the other and
the effects of these factors on the dependent variable are known as crossover effects.
1.5.6
Estimation Methods
Standard methods of estimation in mixed models with a normal response are
maximum likelihood (ML) and restricted maximum likelihood (REML). The linear
mixed effects model is as follows:
y = Xβ þ Zb þ ε
The variance–covariance matrix V for a one-way analysis of variance (ANOVA)
with a randomized block effect and with six observations is equal to:
28
1
Elements of Generalized Linear Mixed Models
0
σ 2b
σ 2 þ σ 2b
2
2
σb
σ þ σ 2b 2 0 2
σ þ σb
0
0
σ 2b
0
0
0
0
0
0
0
0
V = VarðyÞ = ZGZ ′ þ σ 2 I =
0
0
σ 2b
σ 2 þ σ 2b
0
0
0
0
0
0
2
σ þ σ 2b
σ 2b
0
0
0
0
σ 2b
σ 2 þ σ 2b
The variance of y11 is V 11 = σ 2 þ σ 2b and the covariance between y11 and y21 is
V 12 = V 21 = σ 2b . These two observations come from the same block. The covariance
between y11 and other observations is zero. In matrix V, all possible covariances can
be found.
1.5.6.1
Maximum Likelihood
The likelihood function l is a function of the observations and the model parameters.
It gives us a measure of the probability of looking at a particular observation y, given
a set of model parameters β and b. The likelihood function for y j b and b for a mixed
model is given by:
lðyjbÞ = -
n
1
1
logð2π Þ - logjRj - ðy - Xβ - ZbÞ0 R - 1 ðy - Xβ - ZbÞ
2
2
2
and
l ð bÞ = -
1
1
Nb
logð2π Þ - logjGj - bT G - 1 b
2
2
2
where Nb represents the total number of random effect levels. Therefore, the joint
distribution of y and b is equal to:
lðy, bÞ = -
1
1 T -1
ðy - Xβ - ZbÞT R - 1 ðy - Xβ - ZbÞ b G b
2
2
Now, after deriving the above expression with respect to β and b and then setting
it to zero and solving the resulting equations with respect to β and b, the maximum
likelihood estimators are obtained:
∂lðy, bÞ
T
∂β
∂lðy, bÞ
∂b
T
= XT R - 1 y - X T R - 1 Xβ - ZT R - 1 Xb
= ZT R - 1 y - X T R - 1 Zβ - ZT R - 1 Zb
Setting them to zero and solving for β and b, we obtain the following linear mixed
equations:
1.5
Mixed Models
29
XT R - 1 X
ZR - 1 X
XT R - 1 Z
ZT R - 1 ZþG - 1
β
b
=
XT R - 1 y
ZT R - 1 y
The solution can be written as:
β
b
=
XT R - 1 X
ZR - 1 X
XT R - 1 Z
ZT R - 1 ZþG - 1
-1
XT R - 1 y
ZT R - 1 y
Here, β is the vector of fixed effects parameters and b is the vector of random
effects parameters. The information of these parameters is related to the two covariance matrices G and R, and it no longer depends on V as in the previous solution.
Moreover, this solution, which is known as Henderson’s (1950) mixed model
equations, is computationally much more efficient than the previous one given for
the parameters (β and b) since it does not need to obtain the inverse of the matrix
V = ZGZ′ + R. The solution to these mixed model linear equations is based on the
assumption that we know the components of G and R, which, in practice, need to be
estimated. Therefore, the following is a popular method for estimating the variance
components of G and R, which is extremely versatile and powerful.
1.5.6.2
Restricted Maximum Likelihood Estimation
The restricted maximum likelihood method is also known as the residual maximum
likelihood method and is extremely useful, among other things, for estimating
variance components. This method is also based on the maximum likelihood
method, but, instead of maximizing the likelihood function of the original data, it
maximizes the likelihood function over a set of errors obtained by removing the
variables from the original response to fixed effects, which are assumed to be known.
That is, now instead of maximizing over y is maximized over Ky but to obtain the
variance components, it is assumed that K is a matrix of constants, such that KX = 0,
which implies that:
EðKyÞ = ðKXβ þ KZb þ KεÞ = 0
VarðKyÞ = K T VK
This implies that Ky is distributed over N(0, KTVK) and the likelihood of Ky is
called the restricted maximum likelihood (REML). There are many options to
choose K and typically K = I - X(XTX)-1XT, which is the ordinary least squares
residual operator used. Therefore, the log likelihood of Ky is equal to
lðVjKyÞ = -
0
1
n-p
1
logð2π Þ - log K T VK - yT K T K T VK - 1 ðKyÞ
2
2
2
This log likelihood after some algebra, according to Stroup (2012), is equal to:
30
1
lðVjKyÞ = -
Elements of Generalized Linear Mixed Models
n-p
1
1
1
logð2π Þ - logjV j - log X T V - 1 X - rT Vr
2
2
2
2
-
where p = rank (X) and r = y - XβML , where βML = X0 V - 1 X X 0 V - 1 y
The variance components of G and R are estimated with iterative methods such as
the Newton–Raphson or Fisher’s scoring method, which maximizes the likelihood
function l(V| Ky) with respect to the variance components. The maximization process
starts with starting values for the variance components to estimate G and R, and, with
these values of G and R, it is possible to estimate a new, more refined version of the
parameters β and b; then, these values are used to update the estimates of the
variance components of the matrices G and R, and this process continues until the
established convergence is met.
1.5.7
One-Way Random Effects Model
Suppose that we randomly select a possible levels from a sufficiently large set of
levels of the factor of interest. In this case, we say that the factor is random. Random
factors are usually categorical. Continuous covariates that cannot be measured at
random levels are generally known as “systematic” or “fixed” effects (e.g., linear,
quadratic, or even exponential terms). Random effects are not systematic. Let us
assume a simple one-way model:
yij = μ þ τi þ εij ;
i = 1, 2, ⋯, a;
j = 1, 2, ⋯, ni
However, in this case, the treatment effects and the error term are random
variables, i.e., τi N 0, σ 2τ and εij~N(0, σ 2), respectively. The terms τi and εij
are uncorrelated, commonly referred to as “variance components.”
There can be some confusion about the differences between noise factors and
random factors. Noise factors can be fixed or random.
Factors are random when we think of them as being/coming from a random
sample of a larger population, and their effect is not systematic. It is not always clear
when a factor is random. For example, suppose that the vice president of a chain of
stores is interested in the effects of implementing a management policy in his stores
and the experiment includes all five existing stores, he might consider “the store” as a
fixed factor because the levels of the factor “store” do not come from a random
sample. However, if the store chain has 100 stores and takes 5 stores for the
experiment, as the company is considering rapid expansion and plans to implement
the selected new policy at the new locations, then “store” could be considered as a
random factor.
In fixed effects models, the researcher’s interest would focus on testing the
equality of means of treatments (stores). This would not be appropriate, however,
for the case in which 5 stores are randomly selected out of 100 because the
1.5
Mixed Models
31
treatments are randomly selected and we are interested in the population of treatments (stores), not in a particular store or group of stores. The appropriate hypothesis
test for this random effect model would be
H 0 : σ 2τ = 0 vs H a : σ 2τ > 0
Partitioning a standard analysis of variance from the total sum of squares still
works; however, the form of the appropriate test statistic depends on the expected
mean squares. In this case, the appropriate test statistic would be
Fc =
Mean SaquareTreatments
,
Mean SquareError
Fc follows an F-distribution (Fisher–Snedecor) with degrees of freedom a - 1 in the
numerator and N - a in the denominator, where N =
a
i=1
ni .
In a completely random model, we are interested in estimating the variance
components. σ 2τ and σ 2 . To do so, we use the analysis of variance method, which
consists of equating the expected mean squares with the observed values as follows:
σ 2 þ nσ 2τ = Mean SquareTreatments
where σ 2 = Mean SquareError
σ 2τ =
1.5.8
Mean SquareError - σ 2
n
Analysis of Variance Model of a Randomized
Block Design
Consider a one-way analysis of variance model with a randomized block additive
effect. Assume two treatments and three blocks,
yij = μ þ τi þ bj þ Eij
where bj N 0, σ 2b and Eij N ð0, σ 2 Þ with i = 1, 2, 3 and j = 1, 2, 3. The
random effects bj and Eij are independent and uncorrelated. In addition, treatment
effects are assumed to be fixed. The matrix notation of this model is as follows:
32
1
Elements of Generalized Linear Mixed Models
Sistematic
Ramdom
y11
1
1
0
1
0
0
y21
1
0
1
1
0
0
y12
1
1
0
0
1
0
1
0
1
0
1
0
y13
1
1
0
0
0
1
y23
1
0
1
0
0
1
y22
y
=
X
μ
τ1
τ2
β
þ
ε11
ε21
b1
b2
b3
b
þ
ε12
ε22
ε13
ε23
Z
ε
where b~N(0, G) and ε~N(0, R). The variance–covariance matrix G for the random
effects in this case is a diagonal matrix 3 × 3 with diagonal elements σ 2b . Note how
the matrix representation of this model exactly corresponds to the mixed model
formulation. That is,
y = Xβ þ Zb þ ε, where b N ð0, GÞ and ε N ð0, RÞ:
Example An animal nutritionist is interested in comparing the effect of three diets
on weight gain in piglets. To conduct the experiment, the nutritionist randomly
selects 3 litters from a set of 20, each containing 3 healthy, similar-sized, recently
weaned piglets. In each litter, three piglets are selected and each piglet is randomly
assigned to a treatment.
A randomized complete block design (RCBD) is a variation of the completely
randomized design (CRD). In this design, blocks of experimental units are chosen in
such a way that the units within the blocks are as homogeneous as possible with
respect to each other (homogeneous) and different between blocks. In a randomized
complete block design, generally in each block, there is one experimental unit for
each treatment, but this does not limit having more than one experimental unit for
each treatment in each block.
An RCBD has two sources of variation: the factor of interest that includes the
treatments to be studied and the “block factor” that identifies the litters used in the
experiment.
Assumptions in RCBD:
1. Sampling: Blocks (litters) are independently randomly selected and treatments
are randomly assigned to each of the experimental units within each block.
2. Errors are normal, independent, and identically normally distributed with a zero
mean and a constant variance σ 2.
Table 1.23 lists the weight in kilograms of piglets from three different litters
under three different diets. To make inferences about the pattern of weight gain for
the entire population (all litters) of piglets, the litters must be considered in the model
as a random effect. Thus, the linear mixed model describing the variability of piglet
weight gain in this research, as a function of diets, is as follows:
1.5
Mixed Models
33
Table 1.23 Weight gain
(kilograms) of the three litters
of piglets
Litter
1
Table 1.24 Analysis of variance of the randomized complete block design
Sources of variation
Blocks
Diet
Error
Total
Diet1
54.3
53.6
55.2
Diet2
53.1
52.4
57.1
Diet3
59.7
59.7
67.2
Degrees of freedom
b-1=3-1=2
t-1=3-1=2
(t - 1)(b - 1) = 4
tb - 1 = 8
yij = μ þ τi þ bj þ εij for i = 1, 2, 3; j = 1, 2, 3
where yij is the weight observed in the ijth piglet, μ is the overall mean, τi is the fixed
effect due to ith diet, bj is the random effect due to the jth block (litter) assuming
bj N 0, σ 2b , and εij is the independent and identically distributed, approximately
normal, observed error term with mean 0 and variance σ 2, i.e., εij~N(0, σ 2).
Random effects, bj and εij, are assumed to be independent and uncorrelated.
Table 1.24 shows an outline of the analysis of variance for this dataset.
The SAS program to analyze this dataset is as follows:
proc glimmix data=piglets;
class litter diet;
model gain=diet/ddfm=satterthwaite;
random litter;
lsmeans diet/lines;
contrast “Diet1 vs Diet2” diet 1 -1 0;
contrast “Diet2 vs Diet3” diet 1 0 -1;
run; quit;
In the previous syntax, we can mention two commands of great importance in this
example: (1) the “ddfm = satterthwaite” command allows to make a correction of
the degrees of freedom, and this correction is of great importance when the number
of experimental units (UE) is different in each one of the treatments and (2) the
command “lines” serve to obtain the means of “lsmeans” but are grouped with
letters, and, if these averages appear with different letters, then they reflect significant differences.
The output for this code is shown in Table 1.25. Subsection (a) of this table shows
the estimated variance due to litter ðσ 2litter = 5:3117Þ and the mean squared error
σ 2 = 3:2961 . The analysis of variance, part (b), shows that there is a highly
significant effect of diet on piglet weight gain (P = 0.0091). In the results (part c),
we also observe the estimated means and its standard errors (obtained with “lsmeans
diet/lines”) and the grouping of means that are statistically different (part d). In these
last results, we can observe that the weight gain of piglets under treatments I and II
34
1
Elements of Generalized Linear Mixed Models
Table 1.25 Results of the
analysis of variance of the
three different diets tested on
piglet weight gain
(a) Covariance parameter estimates
Cov Parm
Estimate
Standard error
Litter
5.3117
6.4573
Residual
3.2961
2.3307
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Diet
19.02
0.0091
(c) Dietary least squares means
Diet Estimate Standard error DF
t-value Pr > |t|
I
54.3667 1.6939
3.406 32.10
<0.0001
II
54.2000 1.6939
3.406 32.00
<0.0001
62.2000 1.6939
3.406 36.72
<0.0001
III
(d) T grouping of the dietary least squares means (α = 0.05)
LS means with the same letter are not significantly different
Diet
Estimate
III
62.2000
A
I
54.3667
B
II
54.2000
B
Table 1.26 Analysis of
variance under a CRD
Type III tests of fixed effects
Effect
Num DF
Den DF
Diet
F-value
7.28
Pr > F
0.0248
are not statistically different from each other, but they are statistically different with
respect to treatment III.
Since the researcher wishes to make an inference about the entire population of
litters, the factor “litter” must be entered as a random effect; otherwise, the ability of
the F-test to detect differences between treatments is diminished because the P-value
changes from 0.0091 to 0.0248. Another way to see the importance of including
random effects in an ANOVA is to calculate the relative efficiency (RE) between the
two models.
Table 1.26 shows the results of the analysis of variance under a completely
randomized design (CRD), i.e., yij = μ + litteri + eij is as follows:
In this case, if the experiment had been analyzed under a CRD, then the relative
efficiency (RE) between an RCBD and a CRD would be:
CMECRD
=
RE =
CMERCBD
ðSSBRCBD þSCERCBD Þ
t ðb - 1Þ
CMERCBD
=
ðb - 1ÞMSBRCBD þ bðt - 1ÞCMERCBD
ðbt - 1ÞCMERCBD
where CMEDCA is the mean squared error under a CRD, CMERCBD is the mean
squared error under an RCBD, SSBDBCA is the sum of squares due to blocks in an
RCBD, SSEDBCA is the sum of squares of errors in an RCBD, MSBDBCA is the mean
1.6
Exercises
35
Table 1.27 Fit statistics of a
CRD and RCBD
Fit statistics
-2 Res log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson’s chi-square
Pearson’s chi-square / DF
CRD
33.24
41.24
81.24
40.41
44.41
37.90
51.65
8.61
RCBD
31.01
35.01
39.01
33.20
35.20
31.38
19.78
3.30
square due to blocks, and t and b are the number of treatments and blocks,
respectively. If blocks are not useful, then the RE would be equal to 1. The higher
the RE, the more effective the blocking is in reducing the error variance. This value
can be interpreted as the relationship r=b, where r is the number of experimental units
that would have to be assigned to each treatment if a CRD were used instead of
an RCBD.
In Table 1.27, we can observe the mean squared error (MSE) of a CRD and
RCBD (Pearson’s chi-square / DF) obtained with the GLIMMIX procedure in SAS
as well as a series of fit statistics.
The MSE for a CRD and an RCBD are 8.61 and 3.3, respectively. Substituting
these values into the above equation, we obtain
ER =
CMECRD
8:61
=
= 2:609:
CMERCBD
3:3
This value indicates that, an RCBD is 2.609 times more efficient than a CRD. In
other words, this implies that it should have taken, at least, 8 (2.609 × 3 ≈ 8) more
experimental units × treatment units in a CRD to obtain the same MSE as that
obtained in an RCBD.
1.6
Exercises
Exercise 1.6.1 The following dataset corresponds to the growth of pea plants, in
eye units, in tissue culture with auxins ( 0.114 mm). The purpose of this experiment
was to test the effects of the addition of various types of sugars to the culture medium
on growth in length. Pea plants were randomly assigned to one of five treatments:
control (no sugar), 2 % of glucose, 2 % of fructose, 1 % of glucose + 1 % of fructose,
and 2% sucrose. A total of 10 observations were taken in each of the treatments,
assuming that the measurements are approximately normally distributed with constant variance. Here, the individual plants to which the treatments were applied are
the experimental units. The data from this experiment are shown below (Table 1.28):
36
1
Elements of Generalized Linear Mixed Models
Table 1.28 Growth of pea plants in the culture medium with auxins with different types of sugars
Plant
1
2
3
4
5
6
7
8
9
10
Control
75
67
70
75
65
71
67
67
76
68
2% Glucose
57
58
60
59
62
60
60
57
59
61
2% Fructose
58
61
56
58
57
56
61
60
57
58
Table 1.29 Growth (height
in centimeters) of the two
forage species with three
types of fertilizers plus
a control
Species A
Species B
1% Glucose +1% fructose
58
59
58
61
57
56
58
57
57
59
Fertilizer
Control
21
19.5
22.5
21.5
20.5
21
23.7
23.8
23.8
23.7
22.8
24.4
F1
32
30.5
25
27.5
28
28.6
30.1
28.9
30.9
34.4
32.7
32.7
F2
22.5
26
28
27
26.5
25.2
30.6
30.6
28.1
34.9
30.1
25.5
2% Sucrose
62
66
65
63
64
62
65
62
62
67
F3
28
27.5
31
29.5
30
29.2
36.1
36.1
38.7
37.1
36.8
37.1
(a) Write the statistical model that best describes this dataset, indicating its
components.
(b) Calculate the analysis of variance for this experiment.
(c) Is there any significant difference between treatments on average plant growth?
Exercise 1.6.2 A forage company wants to test three different types of fertilizers
(F1, F2, and F3) for the production of two forage species (A and B) for cattle and
compare them with a fertilizer they usually apply, which we will call control. For
this, he decides to use 48 pots with 6 replications in the greenhouse to test the
combinations of fertilizers and forage species. The data from this experiment are
shown in Table 1.29:
1.6
Exercises
37
(a) Write and describe the statistical model of the experimental design with all its
components.
(b) Calculate the analysis of variance for this experiment.
(c) Is there any significant difference between treatments on average plant growth?
Exercise 1.6.3 The data in this experiment are the number of plants regrown after
grazing with sheep–goats. The initial size of the plant at the top of its rootstock is
recorded, and the weight of seeds (g) that it produces at the end of the season is the
response or dependent variable. The data for this experiment are as follows
(Table 1.30):
(a) List and describe all the components of the linear mixed model.
(b) Calculate the ANOVA for this dataset and answer the following questions:
Is seed weight influenced by the type of grazing?
Is seed weight influenced by the plant size?
Is the effect of grazing type on plant size influenced by the initial plant size?
Exercise 1.6.4 An experiment was conducted to study the effect of supplementation
of weaned lambs on health and growth rate when exposed to helminthiasis. A total of
16 Dorper (breed 1) and 16 Red Maasai (breed 2) lambs were treated with an
anthelmintic at 3 months of age (after weaning) and randomly allocated into
“blocks” of 4 per breed, classified on the basis of 3-month body weight for
supplemented and unsupplemented groups. Therefore, two lambs in each block
were randomly allocated to supplemented (night-fed cotton seed meal and wheat
bran) and unsupplemented groups. All lambs were kept on grazing for a further
3 months. Data recorded included the initial body weight (kilograms) at weaning and
weight at 3 months after weaning, percentage red blood cell volume (RBCV), and
fecal egg count (FEC) at 6 months of age. Data from this experiment are shown
below (Table 1.31):
(a) List and describe all the components of the linear mixed model.
(b) Calculate the ANOVA for this dataset and answer the following questions:
Did supplementation improve weight gain? Did supplementation affect PRBC
and FEC, and were there differences in weight gain, PRBC, or FEC between breeds?
38
Table 1.30 Fruit production
after grazing
1
Size
6.225
6.487
4.919
5.13
5.417
5.359
7.614
6.352
4.975
6.93
6.248
5.451
6.013
5.928
6.264
7.181
7.001
4.426
7.302
5.836
10.253
6.958
8.001
9.039
8.91
6.106
7.691
8.988
8.975
9.844
8.508
7.354
8.643
7.916
9.351
7.066
8.158
7.382
8.515
8.53
Elements of Generalized Linear Mixed Models
Fruit
59.77
60.98
14.73
19.28
34.25
35.53
87.73
63.21
24.25
64.34
52.92
32.35
53.61
54.86
64.81
73.24
80.64
18.89
75.49
46.73
116.05
38.94
60.77
84.37
70.11
14.95
70.7
80.31
82.35
105.07
73.79
50.08
78.28
41.48
98.47
40.15
52.26
46.64
71.01
83.03
Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
No Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
Grazing
1.6
Exercises
39
Table 1.31 Supplementation trial in Dorper (breed 1) and Red Maasai (breed 2) lambs
Id
349
326
393
71
271
382
85
176
286
183
21
122
374
32
282
94
127
216
133
249
123
222
290
148
142
154
166
322
156
161
321
324
Race
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Sex
2
2
1
1
1
2
2
2
2
1
2
1
1
2
2
1
2
2
1
1
2
2
2
1
2
2
1
1
1
2
1
1
Supplement
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
Block
1
1
2
2
3
3
4
4
1
1
2
2
3
3
4
4
1
1
2
2
3
3
4
4
1
1
2
2
3
3
4
4
IW
8
9
12
12.3
13
15.5
16.3
15.9
11
9.9
11.6
12.5
14.6
14.2
16.3
16.7
7.5
8.2
10.1
8.8
1.6
11.3
12.3
13.1
8.2
9.5
9.7
8.6
10.2
11.2
12.1
13.8
FW
8.9
10.1
12.6
14.6
13.7
16.8
18.2
17.7
13.6
11.7
13.1
14.8
17.9
16.9
20.2
17.7
8.1
9.3
11.7
10.4
12.6
13.5
14.3
14.9
11.5
12.2
12.8
12
13
14.6
15.9
18.1
PRBC
10
11
22
15
19
24
27
21
21
21
25
25
19
22
20
13
26
19
30
28
23
24
22
26
25
35
29
26
28
22
25
24
FEC
6500
2650
750
5200
4800
2450
200
3000
1600
450
2900
300
2250
2800
750
5600
1350
1150
200
0
600
1500
1950
500
850
700
400
800
1550
550
1250
1100
WG
0.9
1.1
0.6
2.3
0.7
1.3
1.9
1.8
2.6
1.8
1.5
2.3
3.3
2.7
3.9
1
0.6
1.1
1.6
1.6
1
2.2
2
1.8
3.3
3.7
3.1
3.4
2.8
3.4
3.8
4.3
IW initial weight, FW final weight, PRBC percentage of red blood cells, FEC fecal egg count, WG
weight gain
40
1
Elements of Generalized Linear Mixed Models
Appendix
Population
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
St. Croix
Plant
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
29
30
31
32
33
34
35
36
37
38
39
Stamens
30.75
33.83
35.67
35.40
33.50
37.40
33.57
29.86
33.80
31.60
32.57
31.80
35.25
30.00
30.50
32.20
32.40
38.50
37.00
33.00
31.40
31.80
30.40
35.20
27.80
31.29
32.83
31.20
33.00
33.80
32.22
32.91
34.50
28.33
30.71
33.00
31.00
35.00
Eggs
13.75
16.17
16.33
17.40
23.50
18.40
21.29
28.71
29.60
25.80
27.50
24.00
17.75
16.83
18.75
21.40
26.25
17.75
25.83
25.25
25.20
25.60
19.20
22.40
20.80
22.71
22.33
17.40
19.20
22.20
27.63
28.73
15.75
17.33
23.14
24.00
20.50
21.83
Total no. of flowers
8
12
6
14
13
10
25
20
17
14
21
13
8
13
9
13
12
8
16
8
15
14
15
22
10
14
20
14
13
13
31
18
9
8
14
14
4
15
Ratio (stamens/ovules)
2.24
2.09
2.18
2.03
1.43
2.03
1.58
1.04
1.14
1.22
1.18
1.33
1.99
1.78
1.63
1.50
1.23
2.17
1.43
1.31
1.25
1.24
1.58
1.57
1.34
1.38
1.47
1.79
1.72
1.52
1.17
1.15
2.19
1.63
1.33
1.38
1.51
1.60
(continued)
Appendix
Population
St. Croix
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
Cedar Creek
41
Plant
40
1
2
3
4
5
6
7
8
9
10
11
15
16
17
18
19
20
21
23
25
27
28
32
33
34
35
36
37
38
40
41
42
43
44
45
46
47
48
49
50
Stamens
35.00
30.17
32.43
28.00
29.22
36.00
30.83
31.75
29.25
32.78
32.67
31.43
33.50
32.83
35.00
33.17
33.29
35.50
35.71
31.38
28.25
31.82
35.13
33.75
32.00
36.29
28.60
33.00
34.90
34.80
30.00
34.50
37.75
33.50
33.00
35.50
32.50
32.67
35.75
31.38
33.83
Eggs
18.00
18.67
15.14
14.00
16.89
17.14
20.17
18.00
19.00
24.44
22.83
21.00
29.50
15.17
15.00
13.83
27.14
19.83
18.86
25.63
17.50
24.91
26.88
21.63
20.80
17.00
16.40
20.80
25.11
19.60
21.17
20.50
29.00
20.75
22.40
21.50
22.00
16.67
21.50
22.88
20.50
Total no. of flowers
10
16
23
15
35
20
15
18
8
24
17
28
4
20
9
15
28
16
21
5
11
37
23
26
14
18
11
14
49
18
16
16
18
10
12
16
14
8
26
22
17
Data: Larkspur plants from two populations in the state of Minnesota
Ratio (stamens/ovules)
1.94
1.62
2.14
2.00
1.73
2.10
1.53
1.76
1.54
1.34
1.43
1.50
1.14
2.16
2.33
2.40
1.23
1.79
1.89
1.22
1.61
1.28
1.31
1.56
1.54
2.13
1.74
1.59
1.39
1.78
1.42
1.68
1.30
1.61
1.47
1.65
1.48
1.96
1.66
1.37
1.65
42
1
Elements of Generalized Linear Mixed Models
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0
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adaptation, distribution and reproduction in any medium or format, as long as you give appropriate
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the copyright holder.
Chapter 2
Generalized Linear Models
2.1
Introduction
In the generalized linear model (GLM) (which is not highly general) y = Xβ + e, the
response variables are normally distributed, with constant variance across the values
of all the predictor variables, and are linear functions of the predictor variables.
Transformations of data are used to try to force the data into a normal linear
regression model or to find a non-normal-type response variable transformation
(discrete, categorical, positive continuous scale, etc.) that is linearly related to the
predictor variables; however, this is no longer necessary. Instead of using a normal
distribution, a positively skewed distribution with values that are positive real
numbers can be selected. Generalized linear models (GLMs) go beyond linear
mixed models, taking into account that the response variables are not of continuous
scale (not normally distributed), GLMs are heteroscedastic, and there is a linear
relationship between the mean of the response variable and the predictor or explanatory variables.
Nelder and Wedderburn (1972) implemented a unified methodology for linear
models, thus opening a window for researchers to design models that can explain the
variation of the phenomenon under study. Later, McCullagh and Nelder (1989)
proposed an extension of linear models, called generalized linear models (GLMs).
They pointed out that the key elements of a classical linear model are as follows:
(i) the observations are independent, (ii) the mean of the observation is a linear
function of some covariates, and (iii) the variance of the observation is a constant. To
further extend these, points (ii) and (iii) are modified as follows: (ii’) the mean of the
observation is associated with a linear function of some covariates via a link function
and (iii’) the variance of the observation is a function of the mean. For more details,
see the study by McCullagh and Nelder (1989). GLMs can be adapted to a wide
variety of response variables. Special cases of GLMs include not only regression and
analysis of variance (ANOVA) but also logistic regression, probit models, Poisson
regression, log-linear models, and many more.
© The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8_2
43
44
2.2
2
Generalized Linear Models
Components of a GLM
The construction of a GLM begins with choosing the distribution of the response
variable, the predictor or explanatory variables to include in the systematic component, and how to connect the mean of the response to the systematic component. The
three important components are described in the following sections:
2.2.1
The Random Component
The first component to specify is the random component, which consists of choosing
a probability distribution for the response variable. This can be any member of the
exponential family of distributions, such as normal, binomial, Poisson, gamma, and
so on.
2.2.2
The Systematic Component
The second component of a GLM is the systematic component or linear predictor,
which consists of a linear combination of explanatory variables (the predictor). The
systematic component of a model is the fixed structural part of the model that
explains the systematic variability between means. The linear predictor is found on
the right-hand side of the equation in the specification of a linear or nonlinear
regression model. Let x1, x2, ⋯, xp be the numerical (dummy) or discrete (category)
predictor (explanatory) variables, then the linear predictor is
ηi = β0 þ β1 x1i þ β2 x2i þ ⋯ þ βp xpi = xTi β
where βT = (β0, β1, β2, ⋯, βp) is the vector of regression parameters
and xTi = 1, x1i , x2i , ⋯, xpi is the vector of predictor variables. Although η is a
linear function, the x’s can be nonlinear in form. For example, η can be a quadratic,
cubic, or higher-order polynomial. The expected value of yi and the linear predictor
ηi are related through the link function. For example, in a Poisson GLM, the
predictor is equal to logðλi Þ = xTi β, since the link is a natural logarithm, better
known as the link log.
In normal linear regression models, the focus is on η and finding the predictors or
explanatory variables that best explain or predict the mean of the response variable.
This is also important in a GLM. Problems such as multicollinearity in normal linear
regression are also problems in generalized linear models.
2.2
Components of a GLM
2.2.3
45
Predictor’s Link Function η
Finally, we will look at the specification of the link function that maps the mean of
the response variable to the linear predictor. The link function allows a nonlinear
relationship between the mean of the response variable and the linear predictor, and
this link g () connects the mean of the response variable with the linear predictor.
That is,
gðμÞ = η
The function must be monotonous (and differentiable). The mean is equal, in turn,
to the inverse transformation of g (), that is,
μ = g - 1 ðηÞ
The most natural and meaningful way to interpret the model parameters is in
terms of the scale of the data. In other words,
μ = g - 1 ðηÞ = g - 1 β0 þ β1 x1i þ β2 x2i þ ⋯ þ βp xpi
It is important to note that the link relates the mean of the response to the linear
predictor and that this is different from transforming directly to the response
variable. If the response variables are transformed (i.e., y’s), then a distribution
must be selected, which describes the population distribution of the transformed
data, thus making the original interpretation of the data more difficult. A transformation of the mean is generally not equal to the mean of the transformed values, that
is, g(E[y]) ≠ E(g[y]). For example, suppose we have a distribution with the following
values (and probabilities):
yi
prob(Y = yi)
1
0.125
2
0.375
3
0.375
4
0.125
The mean of this distribution is E[y] = 1 × 0.125 + 2 × 0.375 + 3 × 0.375 + 4 ×
0.125 = 2.5. Therefore, the logarithm of the mean of this distribution is ln(E[y])
= ln (2.5) = 0.916, whereas the mean of the logarithm is equal to E(ln[y]) = 0.845.
The value of the linear predictor η could potentially equal any value, but the expected
values of the response variable – as in the case of counts or proportions – can be
bounded. If there are no restrictions on the response variable (positive or negative
real numbers), then the “identity link” function could be used, where the mean is
identical to the linear predictor, that is,
μ = η:
46
2
Generalized Linear Models
As mentioned before, the link function establishes a connection between the
linear predictor η and the mean of the distribution μ. It is important to note that the
link function in some cases is in a sense similar to a function transformation, in that it
establishes only a mathematical connection between the parameters of the model. A
function transformation is applied to the observations to better understand the
relationship between the mean and the response variables or, in some cases, to
stabilize the variance. Special cases are mentioned below:
(a) For a normal distribution, the link function is the identity function, η = μ, the
variance function is constant. i.e., Var(μ) = 1, and the scale parameter is the
variance, i.e., ϕ = σ 2, allowing the use of ordinary least squares in parameter
estimation in procedures such as linear regression, analysis of variance
(ANOVA) models, or analysis of covariance (ANCOVA) models.
(b) In a binomial distribution, the response variable takes binary values like 0 and
1 or represents the relative frequency, i.e., yi = ei/ni, where ei is the number of
successes and ni is the number of trials. The mean is a probability (π) and
therefore must be between 0 and 1. The linear predictor is not bounded.
Therefore, the link function must map the real line within the interval [0, 1]. A
natural link function for binomial data is the logit link:
η = log
π
eη
→ π=
ð1 - π Þ
ð 1 þ eη Þ
Another useful alternative for these types of data is the probit link function:
η = Φ - 1 ðπ Þ → π = ΦðηÞ
where Φ is the cumulative distribution function of a standard normal distribution.
The variance of the function has the form Var(π) = (π/(1 - π)) and the scale
parameter ϕ is known and is equal to 1 (ϕ = 1). The difference between the logit
and probit estimators is important if the estimated probabilities are extremely
small or extremely close to 1, indicating that large sample sizes are required for an
effective inference. Both the logit and probit functions produce extremely close or
equivalent results, especially with probability values around 0.5.
(c) For a Poisson distribution, the link function is the natural log:
η = logðλÞ → λ = eη
The variance of the function has the form Var(λ) = λ, and, similar to the binomial
distribution, the scale parameter is 1. Poisson models with a log link function are
often referred to as log-linear models, commonly used when there are contingency (data frequency) tables with at least two entries.
2.2
Components of a GLM
47
(d) A gamma distribution has a link function of the form:
η=
1
1
→ μ=
η
μ
The variance of the function is given by Var(μ) = μ2and the scale parameter ϕ is
usually unknown. In some cases, the log link function is commonly used, which
results in an exponential inverse link. It should be noted that the link function
does not map the range of the means contained within the linear predictor.
Therefore, given its limitations, the theory only provides reasonable approximations for most applications. An exponential distribution is a special case of the
gamma distribution.
Previously, the classical methods for working with non-normal data – before the
advances in computational methods – consisted of using direct transformation of the
response variable, that is, the data were transformed using the function t( y) before
being analyzed. The goal of the transformation was to obtain a simple connection
between the mean and the linear predictor. However, obtaining a consistent scale of
variation when selecting a transformation is vitally important. The usual way for
selecting a suitable transformation is based on the assumption that, within the region
of variation of the random variable, the transformation can capture the variability
adequately through a simple linear approximation of the mean. That is, if the random
variable y has a distribution with a mean μ and variance σ 2(μ), we want to find a
transformation t( y) such that it is forced to have a constant variance (stabilizes the
variance). The commonly used functions to stabilize variance are the square root
p
y when data have a Poisson distribution; the arcsine square root when data are
binomial; and the logarithmic transformation for data with a constant coefficient of
variation.
Table 2.1 provides an overview of the most common link functions that will give
admissible values for certain types of response variables and the corresponding
inverse of the link function.
Table 2.1 Common link functions for different response variables
Type of response
Normal
Poisson
Binomial ratio
Media
μ
c
π
Variance
σ2
λ
π(1 - π)/N
Exponential
Gamma
Negative binomial
μ
μ
λ
μ2
λ + λ2c
g(μ) = η
log(λ)
(logit) log (π/1 - π)
(probit) Φ-1(π)
Iog(μ)
(inverse)1/μ
log(λ)
g-1(η)
μ=η
λ = e(η)
π = e(η)/(1 + eη)
π = Φ(η)
μ = e(η)
μ = 1/η
λ = e(η)
Note: Φ is the cumulative distribution function of a standard normal distribution; μ and π are the
expected values of the response; η is the linear predictor; and ϕ is the scale parameter
48
2.3
2
Generalized Linear Models
Assumptions of a GLM
According to McCullagh and Nelder (1989) and Agresti (2013) in Chap. 4, a GLM is
defined under the following assumptions:
(a) The data y1, y2, ⋯, yn are independent.
(b) The response variable yi does not necessarily have to have a normal distribution,
but we usually assume a distribution from an exponential family (e.g., binomial,
Poisson, multinomial, gamma, etc.).
(c) A GLM does not assume a linear relationship between the dependent variable
and the independent variables, but it does assume a linear relationship between
the response transformed in terms of the link function and the explanatory
variables; for example, for logit(π) from a binary logistic regression, logit
(π) = β0 + βx.
(d) The predictor (explanatory) variables may be in terms of power or some other
nonlinear transformations of the original independent variables.
(e) The assumption of homogeneity of variance need not be satisfied. In fact, it is not
possible in many cases, given the structure of the model and the presence of
overdispersion (when the observed variance is larger than what the model
assumes).
(f) Errors are independent but are not normally distributed.
(g) The estimation method is maximum likelihood (ML) or other methods instead of
ordinary least squares (OLS) to estimate the parameters.
2.4
Estimation and Inference of a GLM
Estimators of the regression coefficients for linear models with a normal response are
obtained using least squares or ML, and significance tests are generally used to
compare the sum of least squares under different hypothesis tests using the F-test. It
is worth mentioning that these tests are exact, and, so, no approximations are
required for their implementation.
GLMs offer a natural extension of this situation in the sense that: (1) The
computational calculations used to determine the ML estimations of the regression
parameters/coefficients are highly similar to those used in cases when the response is
normal, with the difference being that the estimation process is iterative, which
produces successive approximations that converge to the ML estimates. (2) In the
inference procedures, the test statistic commonly used is the likelihood ratio test,
which is parallel to the F-tests in linear models with a normal response. Thus, GLMs
provide a uniform method of estimation and inference. Estimation of parameter β is
highly similar to the ML method, whereas the inference methods are generally
approximations since they are based on the theory of the distribution of a sufficiently
large sample, as in the case of the likelihood ratio method. There are several
alternative tests such as the Wald test, test scores, and the likelihood ratio test.
2.5
Specification of a GLM
2.5
49
Specification of a GLM
In the following examples, we will describe the components of a GLM for some
normal, gamma, binomial, and Poisson regression models.
2.5.1
Continuous Normal Response Variable
In simple linear regression models, the expected mean value of a continuous
response variable depends on a set of explanatory variables, as follows:
y i = β 0 þ β 1 xi þ εi ,
εi N 0, σ 2
Equivalently,
Eðyi jxi Þ = β0 þ β1 xi
This GLM can be expressed in terms of its three components:
Distribution: yi N μi , σ 2
E ð yi Þ = μ i
Varðyi Þ = σ 2
Linear predictor : ηi = β0 þ β1 xi
Link function: ηi = μi
ðidentity linkÞ
where β0 and β1 are the intercept and slope, respectively. This means that we are
expressing the linear model as a GLM.
Example 1 A simple linear regression analysis was performed on the diamond price
( y) as a function of the number of carats (Table 2.2) and assuming that the response
variable “y” has a normal distribution with a mean β0 + β1xi and variance σ 2.
The basic Statistical Analysis Software (SAS) syntax for simple linear regression
is as follows:
proc reg ;
model price=weight/clb p r;
output out=diag p=pred r=resid;
id weght;
run;
In the above program, “proc reg” invokes a linear regression procedure in SAS.
The “clb” option generates a confidence interval for the slope and intercept. The “p”
50
2
Generalized Linear Models
Table 2.2 Diamond price (dollars) based on weight (carats)
Weight
0.17
0.16
0.17
0.18
0.25
0.16
0.15
0.19
0.21
0.15
Price
355
328
642
342
322
485
483
Weight
0.18
0.28
0.16
0.2
0.23
0.29
0.12
0.26
0.25
0.27
Price
462
823
336
498
595
860
223
663
Weight
0.18
0.16
0.17
0.16
0.17
0.18
0.17
0.18
0.17
0.15
Price
468
345
352
332
353
438
318
419
346
Weight
0.17
0.32
0.32
0.15
0.16
0.16
0.23
0.23
0.17
0.33
Price
918
919
298
339
338
595
553
345
945
Weight
0.25
0.35
0.18
0.25
0.25
0.15
0.26
0.15
0.43
Price
655
1086
443
678
675
287
693
316
.
Table 2.3 Regression analysis results
β0
Estimated parameters
Effect
Estimate
Intercept -259.63
β1
Weight
3721.02
σ
Scale
1013.82
2
Standard error
17.3189
81.7859
Degree of freedom (DF)
46
46
t- value
-14.99
Pr > |t|
<0.0001
45.50
<0.0001
211.40
option generates fitted values and standard errors. The “r” option performs a residual
analysis (i.e., checks assumptions). The “output out” statement generates a new
dataset called “diag” containing the residuals and the predicted/adjusted values. The
“id weight” statement adds the specified variable to the fitted values output.
Part of the results is shown in Table 2.3. The estimated parameters, obtained from
“proc reg,” are shown below:
Note that the estimated parameters are all statistically significantly different from
zero. Then, the linear predictor takes the form:
η = - 259:63 þ 3721:02 × weighti
If the response variable “y” does not fit the data well, then the normal distribution
may barely represent the response distribution; that is, it would weakly explain the
variability of the data and, consequently, the “identity” may not be the best link
function, since the linear predictor would not include all the relevant information or
some combination of the three components of the GLM. Although other fit measure
statistics exist in the linear regression model, such as the coefficient of determination
(R2), the residual analysis is used to determine whether there is a good fit of the
model or whether the assumptions of a Gaussian model are met. In this example, the
value of R2 is R2 = 0.9783, and this value indicates that the model used explains
97.83% of the total variability of the dataset. In Fig. 2.1, we can see that the simple
linear regression model provides a good fit to this dataset.
2.5
Specification of a GLM
51
Fig. 2.1 A dot plot of price vs. weight (carat) and fitted model
2.5.2
Binary Logistic Regression
Logistic regression and other binomial response models are widely used in research
areas like biological sciences and agriculture. Given their importance in this section,
some relevant features of these models are mentioned.
Let yi be the observed response on a set of p explanatory variables x1, x2, ⋯, xp
whose distribution yi is binomial with ni independent Bernoulli trials and probability
of success π i on each trial, i.e.,
yi Binomialðni , π i Þ
Then, we can model the response using a GLM with a binomial response. The
linear predictor in this case will be equal to
log
πi
= β0 þ β1 x1i þ ⋯ þ βp xpi
1 - πi
commonly known as “logit” because logit is defined as:
logitðπ i Þ = log
πi
ð1 - π i Þ
52
2
Generalized Linear Models
which models the logarithm of the odds ratio, ð1 -πi πi Þ , as a function of the predictor
variables. The components of this GLM for binomial data are:
Distribution: yi Binomialðni , π i Þ, with mean and variance
E ðyi Þ = ni π i and Varðyi Þ = ni π i ð1 - π i Þ
Linear predictor : ηi = β0 þ β1 x1i þ ⋯ þ βp xpi
Link function : ηi = logitðπ i Þ = log
πi
ðlogit linkÞ
ð1 - π i Þ
Another highly useful link function – when you have experiments – is the
“probit” link ηi = Φ-1(π i), which was mentioned before.
The basic GLM for this dataset, under the probit link, is almost identical to the
logit link as seen below:
Distribution: yi Binomialðni , π i Þ
Linear predictor : ηi = β0 þ β1 xi þ ⋯ þ βp xp
Link function : ηi = probitðπ i Þ = Φ - 1 ½π i :
Example 1 An engineer is interested in studying the effect of temperature (Temp)
from 0 to 40 °C and time in days from 0 to 15 days on the germination of seeds of a
certain crop. For this reason, he placed seeds in different pots containing moist soil.
After a certain number of days, the number of germinated seeds was counted. If the
seeds germinated, then y = 1; otherwise, y = 0. The probability of germination π ij
can be modeled through
ηij = β0 þ β1 Dayi þ β2 Tempj
where ηij is the linear predictor and β0, β1, and β2 are the parameters to be estimated.
In this GLM, the link function is
ηij = logit π ij = log
π ij
1 - π ij
and the probability in the interval (0, 1) is computed through the inverse of the link
function
π ij =
1
= g - 1 ηij
1 þ exp ηij
2.5
Specification of a GLM
53
This last expression allows to estimate the probability of germination (π ij) under
different temperature conditions (°C) and time periods (days). Note that the
nonlinear relationship between the result π ij and the linear predictor ηij is modeled
by the inverse of the link function. In this particular case, the link function is the
logit.
ηij = log
π ij
1 - π ij
= g π ij
For the illustration of this example, a set of data was simulated using the values
β0 = 8, β1 = - 0.19, and β2 = - 0.37 in the linear predictor and the inverse of the
linear function by varying the temperature from 0 to 40 °C and time from 0 to
15 days, i.e.,
π^ij =
1
1þe
ð8 - 0:19 × Tempi - 0:37 × Dayj Þ
Part of the simulated data is shown below:
Temp
0
0
0
0
0
.
.
.
40
40
40
40
40
Days
0
0.5
1
1.5
.
.
.
13
13.5
14
14.5
15
Germ
0.000335
0.000403
0.000485
0.000584
0.000703
.
.
.
0.987991
0.989999
0.991674
0.99307
0.994234
The following commands allow us to perform a binomial regression using the
“logit,” “probit,” and linear regression with the “identity” link. It is important to
mention that we denote temperature as t and days as d in the codes used below.
Logit Regression
proc glimmix data=germ;
model p = t d/solution dist=binomial link=logit cl;
output out=logitout pred(noblup ilink)=predicted resid=residual;
run;
54
2
Generalized Linear Models
Probit Regression
proc glimmix data=germ;
model p = t d/solution dist=binomial link=probit cl;
output out=probitout pred(noblup ilink)=predicted resid=residual;
run;
Linear Regression (Identity)
proc glimmix data=germ ;
model p = t d/solution cl dist=normal;
output out=identity out pred(noblup ilink)=predicted resid=residual;
run;
“proc GLIMMIX” in SAS uses complex models without modifying the response
variable as occurs when a direct transformation is applied to the response variable.
Instead, GLIMMIX uses a link function of the response variable that is modeled as
having a linear relationship with the explanatory variables. The “model” command
specifies the response variable p as a function of the explanatory variables t and d,
which define Xβ. The “solution” option in the model specification invokes the
regression procedure to list the fixed effects parameter estimates of the model
(β0, β1, and β2). The “dist” option is used to specify the distribution of the response
variable, and the “link” option is used to specify the link function.
To get predicted probability values for each observation, the “output” option in
proc GLIMMIX is used. Two types of predicted values can be obtained with the
“output” option. The first type is the solution for the random effects (best linear
unbiased predictors (BLUPs)) in the linearized model, and the second type is the
predictions based on the fixed effects (best linear unbiased estimators (BLUEs))
(pred(noblup ilink) = predicted). The “ilink” sub-option in the “pred” option asks for
the inverse function of the predicted values, that is, the probabilities of the predictions that are stored under the predicted file name. Finally, the “resid” option is
used to request the residuals of the regression, which are stored in the residual.
Table 2.4 shows part of the output (analysis of variance (part (a)) and estimation
and significance of fixed effects (part (b)) of the regression procedure using the logit
link function.
Table 2.4 Estimation and
significance of fixed effects
using the logit link function
(a) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
T
1
2508
551.28
D
1
2508
407.19
(b) Parameters estimates
Standard
Effect
Estimate error
DF
t-value
Intercept -8.0000 0.3189
2508 -25.08
T
0.1900 0.008092 2508
23.48
D
0.3700 0.01834
2508
20.18
Pr > F
<0.0001
<0.0001
Pr > |t|
<0.0001
<0.0001
<0.0001
2.5
Specification of a GLM
55
Table 2.5 Parameter estimates, linear predictor, and probability of linear, logit, and probit models
Link function
Linear
Logit
Parameter
β0
β1
0.022
β2
0.0412
β0
β1
Probit
Estimated value
-0.417
-8.00
η
1.149
π
0.873
5.95
0.962
3.362
0.965
0.190
β2
0.370
β0
-4.483
β1
0.106
β2
0.207
*The linear predictor η and the probability π were estimated using D = 15 and T = 30
Fig. 2.2 (a, b) Probability of seed germination as a function of temperature and day
In Table 2.5, parameter estimates of the linear predictor for the generalized linear,
logit, and probit models are presented. The probabilities estimated by the probit and
logit models are almost identical to each other, but those of the linear probability
model are different; this is because the data were generated with a binomial distribution, whereas the estimated linear predictor differs substantially from the linear
predictor under the link probit and logit.
In Fig. 2.2a, b, we observe that in an interval between 3 and 7 days and 0 and
15 °C, there is approximately 20% seed germination, but, while both factors
increase, the germination percentage also increases substantially.
2.5.2.1
Model Diagnosis
For a linear model, a plot of the predicted values against the residuals is probably the
simplest way to decide whether the model used provides a good fit to the data; but,
for a GLM, we must decide on the appropriate scale to use for the fitted values.
56
2
Generalized Linear Models
Fig. 2.3 Predicted vs. residual values using the logit link
Generally, it is better to use linear predictors η in the plot rather than the predicted
responses μ. If there is no linear relationship between the linear predictors and the
residuals, then it could indicate a lack of fit in the model. For a linear model, we
could perform a transformation of the response variable, but this is not highly
recommended for a GLM as this could change the response distribution. Another
alternative would be to change the link function, but since there are not many link
functions that allow interpreting a model easily, this is not a good option. Moreover,
changing the linear predictor or transforming the predictor variables would not be the
best way to go.
Figures 2.3, 2.4, and 2.5 show the linear predictor versus residual (we can also see
the predicted value versus the residual). By investigating the nature of the relationship between the predictors and the residuals in Fig. 2.3, we can see that there is a
linear relationship between the predictor and the residual, using the logit function,
whereas the probit and identity functions do not show this linear relationship.
However, with the probit link function, we observe a curvilinear relationship
between the predictor and the residual, which may be because homogeneity of
variance is not satisfied under this link function. Therefore, the logit link is shown
to be the best choice.
Example 2 Fruit flies can be a year-round problem in fruit-growing areas in many
regions of the world, such as in Mexico, and are most common especially in late
summer and fall because ripe or fermented fruits and vegetables attract insects by
serving as a natural host. If these insects are not controlled, economic losses in fruitgrowing areas could be large and devastating to the producers. In response to this,
entomologists have implemented experiments to help mitigate the damages caused
by these insects. One such experiment attempted to establish the relationship
between the concentration of a toxic agent (nicotine) for 5 hours and the number
2.5
Specification of a GLM
57
Fig. 2.4 Predicted values vs. residuals using the probit link
Fig. 2.5 Predicted vs. residual values using the identity link
of insects killed (common fruit fly); the data are shown in Table 2.6, and, for more
information, see the study by Myers et al. (2002).
The number of dead insects can be modeled under a binomial distribution (n, π).
Let yi denote the number of dead insects at a concentration i. The GLM components
for this dataset are:
58
2
Generalized Linear Models
Table 2.6 Ratio of the concentration of a toxic agent to the number of fruit flies killed
Concentration (g/100 cc)
0.1
0.15
0.20
0.30
0.50
0.70
0.95
Number of insects (n)
47
53
55
52
46
54
52
Dead insects ( y)
8
14
24
32
38
50
50
Proportion of dead insects
0.17
0.264
0.436
0.615
0.826
0.926
0.962
Distribution: yi Binomialðni , π i Þ, with mean and variance
: Eðyi Þ = ni π i and Varðyi Þ = ni π i ð1 - π i Þ
Linear predictor : ηi = β0 þ β1 conci
πi
Link function : ηi = logitðπ i Þ = log ½
ðlogit linkÞ
ð1 - π i Þ
Note that we are using conci to denote the independent variable nicotine toxicant
concentration. The following SAS code allows us to perform a binomial regression
for the fruit fly dataset:
fly data;
input conc n y;
datalines;
0.1 47 8
0.15 53 14
0.2 55 24
0.3 52 32
0.5 46 38
0.7 54 50
0.95 52 50
;
proc glimmix data=nobound fly;
model y/n = conc/dist=binomial link=logit solution;
run;
The above syntax produces the following output:
The analysis of variance (Table 2.7 a) shows that there is a highly significant
effect of nicotine concentration on the number of flies killed (P = 0.0004). From the
results obtained, we can observe that, in part (b), the maximum likelihood estimator
for the intercept and slope are β0 = - 1:7361 and β1 = 6:2954, respectively, which
are used to construct the linear predictor:
ηi = - 1:7361 þ 6:2954 × conci
2.5
Specification of a GLM
Table 2.7 Results of the
analysis of variance with the
logit link
59
(a) Type III tests of fixed effects
Effect
Num DF
Den DF
Conc
1
5
(b) Parameter estimates
Effect
Estimate Standard error
Intercept -1.7361 0.2420
Conc
6.2954 0.7422
F-value
71.94
DF
5
5
t -value
-7.17
8.48
Pr > F
0.0004
Pr > |t|
0.0008
0.0004
Fig. 2.6 Proportion of dead insects as a function of nicotine concentration
Therefore, with the logistic regression model, we can estimate the probability that
an insect dies when exposed to a certain concentration i of nicotine using the
following expression:
π ðconci Þ =
eη i
1 þ eη i
=
e - 1:7361þ6:2954 × conci
1 þ e - 1:7361þ6:2954 × conci
A plot of the mean proportion of dead insects exposed to a certain concentration
of nicotine and the regression curve (linear, quadratic, and cubic) is shown in
Fig. 2.6. In this figure, we observe that as the nicotine concentration increases, the
mean proportion of dead insects increases. The best linear predictor is of a quadratic
order.
2.5.3
Poisson Regression
Often, the outcome of a variable is numerical in the form of counts. Sometimes it is a
count of rare events such as, for example, (1) the number of plants infected by a
60
2 Generalized Linear Models
certain disease in a population over a period of time, (2) the number of insects
surviving after the application of an insecticide over time, (3) the number of dead fish
found per cubic kilometer due to a certain pollutant, (4) the number of sick animals
occurring in a given month in a given country, and so on. The Poisson probability
distribution is perhaps the most widely used for modeling count-type response
variables. As λ (the average count) increases, the Poisson distribution grows symmetrically and eventually approaches a normal distribution.
The Poisson likelihood function is appropriate for nonnegative integer data and
this process assumes that events occur randomly over time, so the following
conditions must be met:
(a) The probability of at least one occurrence of an event in a given time interval is
proportional to the length of the interval.
(b) The probability of two or more occurrences of an event within an extremely
small interval is negligible.
(c) The number of occurrences of an event in disjoint time intervals are mutually
independent.
The probability distribution of a Poisson random variable "y, " which represents
the number of successes occurring in a given time interval or in a given region of
space, is given by the expression
Pðy = kÞ =
e - λ λk
,
k!
λ > 0, k = 1, 2, ⋯
where λ is the average number of successes (the average count) in a time or space
interval. The mean and variance of this distribution are the same, that is,
EðyÞ = VarðyÞ = λ
Poisson regression belongs to a GLM and is appropriate for analyzing count data
or contingency tables. A Poisson regression assumes that the response variable “y”
has a Poisson distribution and that the logarithm of its expected value can be
modeled by a linear combination of unknown parameters and independent variables.
As in a standard linear regression, the predictors, weighted by the coefficients of x1,
x2, ⋯, xp, are summed to form the linear predictor,
P
ηi = β 0 þ
xpi βp
p=1
where β0 is the intercept and βp is the slope of the covariates xp ( p = 1, ⋯, P). Thus,
the expected value of yi and the linear predictor ηi are related through the link
function. The components of a GLM with a Poisson response (yi ~ Poisson(λi)),
where λi is the expected value of yi, are as follows:
2.5
Specification of a GLM
61
Fig. 2.7 Students infected with the disease
Distribution: yi Poissonðλi Þ, with E ðyi Þ = Varðyi Þ = λi
Linear predictor : ηi = β0 þ β1 x1i þ ⋯ þ βp xpi
Link function: ηi = logðλi Þ = gðλi Þ
ðlog linkÞ
Example 1 The following dataset corresponds to the number of students diagnosed
(Fig. 2.7) with a certain infectious disease within a period of days of an initial
outbreak. We will fit a generalized linear model for “count” data assuming a Poisson
distribution.
Note that the response distribution is skewed to the right and that the responses
are positive integers. Since the response variable is count, the initial choice of a
Poisson distribution is reasonable for this dataset with its canonical link, the natural
logarithm. The number of “days elapsed” after the initial disease outbreak is the
predictor variable in the systematic component. Thus, the GLM for this dataset
(Appendix: Data: Infected students) is:
Distribution: Inffected studentsi Poissonðλi Þ
Linear predictor: ηi = βo þ β1 Days
Link function: ηi = logðλi Þ ðlog linkÞ
Part of the data is shown below:
Days elapsed
1
2
3
.
Infected students
6
8
12
.
(continued)
62
2
Generalized Linear Models
Days elapsed
.
.
109
110
112
Infected students
.
.
1
1
0
For the purposes of implementation, we use days to denote elapsed days and
students to denote infected students. We can employ the Poisson regression model
using GLIMMIX in SAS, as shown below:
proc glimmix data=students method=laplace;
model students=days/solution dist=poisson link=log;
output out=sal_infection pred(noblup ilink)=predicted
resid=residual;
run;
The “proc GLIMMIX” statement invokes the SAS generalized linear mixed
model (GLMM) procedure. The “model” command specifies the response variable
and the predictor variable, whereas the “solution” option in the model specification
requests a listing of the fixed effects parameter estimates. The “dist = poisson”
option specifies the distribution of the data, and the “link = log” option declares the
link function to be used in the model. The default estimation technique in generalized linear mixed models is restricted pseudo-likelihood (the “RPSL method”); in
this example, we use “method = laplace.” The “output” option creates a dataset
containing predicted values and diagnostic residuals, calculated after fitting the
model. By default, all variables in the original dataset are included in the output
dataset, whereas the “out = sal_infection” statement specifies the name of the output
dataset. The “pre(noblup ilink) = predicted” option calculates the predicted values
without taking into account the random effects of the model, and “ilink” calculates
the statistics and predicted values at the scale of the data. Finally, the “resid = residual
option” calculates the residuals.
The probability estimation of a GLMM involves an integral, which, in general,
cannot be calculated explicitly. “GLIMMIX,” by default, uses the RSPL method, but
it also offers different options such as the quadrature and Laplace integration
method, among others. These integral approximation methods approximate the
probability function of an GLMM, and the optimization of the function is numerically approximated. These methods provide a real objective function for optimization. For more details, see the SAS manual. However, in a GLM, this approximation
involving the integral is not necessary since an exact solution can be obtained to
estimate the parameters, as there are no random effects. The results of this analysis
are shown below (Table 2.8).
The fit statistics in part (a) (“Fit statistics”) give us an idea of the quality of the
goodness of the fit of the model; these statistics are very useful when we are
proposing different models to try and find the best model for the data. In this case,
the value of the generalized chi-squared statistic divided over its degrees of freedom
2.5
Specification of a GLM
Table 2.8 Results of the
analysis of variance
63
(a) Fit statistics (Akaike’s information criterion (AIC), a small
sample bias corrected Akaike’s information criterion
(AICC), Bozdogan Akaike’s information criterion (CAIC),
Schwarz’s Bayesian information criterion (BIC), Hannan and
Quinn information criterion (HQIC))
-2 Log likelihood
389.11
AIC (smaller is better)
393.11
AICC (smaller is better)
393.22
BIC (smaller is better)
398.49
CAIC (smaller is better)
400.49
HQIC (smaller is better)
395.29
Pearson’s chi-square
84.95
Pearson’s chi-square / DF
0.78
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Days
1
102.28
<0.0001
(c) Parameter estimates
Standard
Effect
Estimate error
DF t-value Pr > |t|
Intercept 1.9902
0.08394
23.71
<0.0001
Days
0.001727
<0.0001
0.01746
10.11
is close to 1. This indicates that the variability of these data has been reasonably
modeled and that there is no residual overdispersion. The value of the generalized
chi-squared statistic divided over its degrees of freedom (Pearson′s chi - square/DF)
is the experimental error of the analysis.
The “Type III tests of fixed effects” (in part (b)) and the solution for the intercept
and the days effect (“Parameter estimates”) in part (c) are shown in Table 2.8. The
negative coefficient of the covariate days indicates that as the number of days
increases, the average number of students diagnosed with the disease decreases.
That is, we reject the null hypothesis (P = 0.0001) that the expected number of
infected students is the same as the number of days increases.
We see that with a 1-day increase in the infection period, the expected
(or average) number of students diagnosed with the disease decreases by a factor
of e-0.01746 = 0.9827.
The estimated linear predictor for this GLM is:
ηi = 1:9902 - 0:01746 × Days
For example, we can calculate the probability of diagnosing "k = 2" infected
students in a period of 2 days; i.e., "Days = 2"as follows:
PðY i = kÞ =
exp
- λi
k!
λi
k
64
2
Generalized Linear Models
Fig. 2.8 Infected students and a Poisson regression fit
exp ð- exp½1:9902 - 0:0146 × 2Þ ðexp½1:9902 - 0:0146 × 2Þ2
2
2
ð- expð1:961ÞÞ
ðexpð1:961ÞÞ
exp
=
= 0:0207
2
PðY i = 2Þ =
This value indicates that the probability of observing/diagnosing two students
with the disease in a 2-day period is 0.0207 (2.0701%).
In Fig. 2.8, we observe that the Poisson model is a good candidate for modeling
this dataset, since there is no overdispersion in this regression model.
Example 2 A forest engineer is interested in modeling the number of trees recently
infected by a certain virus. The data that he has are age (years), height (meters) of the
trees, and the number of infected trees. Using a linear model could result in negative
values of the parameter λ, which would not make sense. The link function g(λ) for a
Poisson error structure is the logarithm. Therefore, the GLM, defining yi = infected
treesi, can be as follows:
Distribution: yi Poissonðλi Þ
Linear predictor : ηi = β0 þ β1 × Agei þ β2 × heighti
Link function: ηi = logðλi Þ = gðλi Þ ðlog linkÞ
For this example, a dataset was simulated using the following parameter values:
β0 = - 2, β1 = - 0.03, and β2 = - 0.04. In addition, in order to obtain the linear
predictor, the variable age (years) varied from 0 to 50 and height (meters) from 0 to
30, both with increments in one unit. Thus, the values of yij were simulated with the
following expression:
2.5
Specification of a GLM
65
Fig. 2.9 (a, b) Probability of tree infection as a function of tree height and age in years
yij = expð - 2 - 0:03 × Agei - 0:04 × Heightj Þ
In Fig 2.9a, b, we can see that at a young age, between 1 and 10 years and at a
height of no more than 10 meters, trees are more susceptible to be infested by the
virus. However, as their age increases, trees show greater resistance.
The following SAS code fits a Poisson regression model with two predictor
variables, assuming that there is no interaction between the two explanatory
variables.
proc glimmix data=infection method=laplace;
model infection=age height /solution dist=poisson link=log;
output out=sal_infection pred(noblup ilink)=predicted
resid=residual;
run;
In Table 2.9 part (a), the analysis of variance shows that age and tree height are
highly significant, indicating that both variables help explain the infection mechanism of the trees through a Poisson model (P < 0.0001).
The linear predictor for this GLM, with Poisson distribution, in the response
variable is:
ηij = - 2 - 0:03 × Agei - 0:04 × Heightj
The estimated values of the parameters of each of the explanatory variables
indicate that as age (years) and height (meters) increase by one unit, the tree is less
susceptible to the virus. If we want to calculate the probability of diagnosing "k = 3”
infected trees with the virus when they are 2 years old and 3-meters tall, we can use
the following equation:
66
2
Generalized Linear Models
Table 2.9 Part of the results of the analysis of variance under a Poisson distribution
(a) Type III tests of fixed effects
Effect
Num DF
Age
1
Height
1
(b) Parameter estimates
Effect
Estimate
Intercept
-2.0000
Age
-0.03000
Height
-0.04000
Den DF
6158
6158
F-value
43.20
29.10
Standard error
0.1388
0.004564
0.007415
PðY i = kÞ =
exp
DF
6158
6158
6158
- λi
λi
t-value
-14.41
-6.57
-5.39
Pr > F
<0.0001
<0.0001
Pr > |t|
<0.0001
<0.0001
<0.0001
k
k!
^ i = 3Þ
PðY
=
expð - exp ½ - 2 - 0:03 × Age - 0:04 × HeightÞ ðexp ½ - 2 - 0:03 × Age - 0:04 × HeightÞ3
3!
=
expð - exp ½ - 2 - 0:03 × 2 - 0:04 × 3Þ ðexp ½ - 2 - 0:03 × 2 - 0:04 × 3Þ3
= 0:000215
3!
This value indicates that the probability of observing/diagnosing three trees
with the virus causing the disease when they are 2 years old and 3-meters tall
is 0.000215 (0.0215%).
A Poisson regression model, sometimes referred to as a log-linear model, is
especially useful when it is used in contingency table modeling. Log-linear models
are models of associations between variables in a contingency table; they treat
variables symmetrically and do not distinguish one variable as a response. They
have a formal structure of double or more entries that can be fitted by binomial or
Poisson regression. These models for contingency tables have several specific
applications in biological and social sciences.
Variables can be nominal or ordinal. A nominal variable has no natural order; for
example, gender (male, female, transgender), eye color (blue, brown, green), and
type of pet (cat, bird, fish, dog, mouse). An ordinal variable has a range of orders; for
example, when you want to measure the degree of consumer satisfaction with the
consumption of a product (very dissatisfied, somewhat dissatisfied, neither satisfied
nor dissatisfied, somewhat satisfied, very satisfied).
2.5
Specification of a GLM
2.5.4
67
Gamma Regression
A gamma distribution is a distribution that occurs naturally in processes for which
waiting times, between events, are relevant. Lifetime data are sometimes modeled
with a gamma distribution. This distribution can take a wide range of forms due to
the relationship between the mean and variance across its two parameters (α and β)
and is suitable for dealing with heteroscedasticity of nonnegative data. The probability of observing a particular value y, given the parameters α and β, is
f ðyÞ =
1
yα - 1 e - ðy=βÞ ; y, α, β > 0
ΓðαÞβα
where Γ(∙) is the gamma function. A gamma regression uses the input variables X’s
and coefficients to make a prediction about the mean of "y, " but it actually focuses
more of its attention on the scale parameter β. The mean and variance of a Gamma
random variable are:
E ðY Þ = αβ = μ and VarðY Þ = αβ2 = μ2 =α
The probability density function gamma can be rewritten in terms of the mean
μ and the scale parameter α as follows:
f ðyÞ =
1
yα
ΓðαÞy μ
α
exp ð- yμÞ ,
α
y>0
Plotting the gamma distribution (Fig. 2.10) with three different values of shape
α = (0.75, 1, and 2), the scale parameter μ has a multiplicative effect. In the gamma
density of the first panel α = 0.75, we see that the density is infinite at 0, whereas in
the second panel α = 1, it corresponds to the exponential density, and, in the third
panel α = 2, we see a skewed distribution.
Fig. 2.10 Gamma density: from left to right, α = 0.75, 1, and 2
68
2
Generalized Linear Models
A gamma distribution can arise in different forms. The sum of "n" independent
and identically distributed exponential random variables with parameter β has a
gamma distribution (n, β). The chi-squared distribution χ 2 is a special case of a
gamma distribution with β = 1/2 and α/2 degrees of freedom.
Theoretically, a Gamma distribution should be the best choice when the response
variable has a real value in the range of zero to infinity and it is appropriate when a
fixed relationship between the mean and variance is suspected. If we expect the
values "y" to be small, then we should expect a small amount of variability in the
observed values. Conversely, if we expect large values of "y, " then we should expect
(observe) a lot of variability.
Models with a gamma distribution with multiplicative covariate effects provide
additional support for modeling nonnegative right-skewed continuous responses,
such as the gamma variable with the log link function. Whether the data are modeled
with an inverse or logarithmic link function will depend on whether the rate of
change or the logarithm of the rate of change is a more meaningful measure. For
example, in studies of yield density that commonly assume that yield per plant is
inversely proportional to plant density (Shinozaki and Kira 1956), the linear
predictor is:
η i = ð β 0 þ β 1 xi Þ - 1
Example 1 In the development of coagulation agents, it is common to perform
in vitro clotting time studies. The following data were reported by McCullagh and
Nelder (1989). Plasma samples from healthy men were diluted to nine different
percentages of prothrombin-free plasma concentration; the greater the dilution, the
more interference with the ability of the blood to clot because the natural clotting
ability of the blood has been weakened. For each sample, clotting was induced by
introducing thromboplastin, a clotting agent, and the time until clotting occurred
(in seconds) was recorded. Five samples were measured at each of the nine concentration percentages, and the mean clotting times were averaged; therefore, the
response is the mean clotting time across the five samples. In Fig. 2.11, the response
variable is plotted against the percentage thromboplastin concentration in which we
observe that the longer clotting times tend to be more variable than the smaller
clotting times, so a linear regression model may not be appropriate.
In this analysis, we will model clotting times as the response variable (yi) with
plasma concentration percentage as the predictor variable. Conc denotes the independent variable concentration. The GLM for this dataset is:
Distribution: yi = Clotting timei Gammaðα, βÞ
Linear predictor: ηi = β0 þ β1 × conci
Link function : μi =
1
ðinverse linkÞ
β0 þ β1 × conci
2.5
Specification of a GLM
69
Fig. 2.11 Clotting time (seconds), depending on the thromboplastin concentration
The following syntax allows us to adjust a GLM with gamma errors in
GLIMMIX:
data coagu;
input num conc y;
datalines;
1 5 118
2 10 58
3 15 42
4 20 35
5 30 27
6 40 25
7 60 21
8 80 19
9 100 18
;
proc glimmix data = coagu;
model y = conc / dist=gamma link=power(-1) solution;
output out=salgamm1 pred(noblup ilink)=predicted resid=residual;
run;
Most of the syntax has already been described in the previous examples; the only
new one is the link = power(-1) option. This statement invokes the inverse
link function in the GLIMMIX procedure.
Some of the output from this analysis is shown in Table 2.10.
The dilution percentage, part (a) in Table 2.10, of the blood plasma concentration
significantly affects the clotting time (P = 0.0004). The values for constructing the
fitted linear predictor are tabulated in part (b) of Table 2.10.
ηi = 0:008686 þ 0:000658 × conci
70
2
Table 2.10 Results of the
regression analysis under a
gamma distribution
Generalized Linear Models
(a) Type III tests of fixed effects
Effect
Num DF
Den DF
Conc
1
(b) Parameter estimates
Effect
Estimate Standard error
Intercept 0.008686 0.002294
Conc
0.000658 0.000103
Scale
0.05213
0.02436
F-value
41.01
DF
.
t-value
3.79
6.40
.
Pr > F
0.0004
Pr > |t|
0.0068
0.0004
.
With the parameterization of the gamma distribution, previously chosen, the
intercept and the beta coefficient corresponding to the concentration variable were
calculated through GLIMMIX in SAS, as well as the scale parameter(α), which in
the SAS output corresponds to the scale. With part of this information, it is possible
to calculate the mean (E[Y] = μ) and variance (Var[Y] = μ2/α) for a concentration
conc = 10 as follows:
y=μ=
1
1
=
= 65:505
0:008686 þ 0:000658 × conc 0:008686 þ 0:000658 × 10
VarðyÞ =
μ2 65:5052
=
= 85818:215
α
0:052
The average time it takes for blood to clot – when a thromboplastin concentration
of 10% is added – is 65.505 seconds with a variance of 85818.215.
2.5.4.1
Model Selection
Selecting a model from a set of candidate models that provides the best fit and largely
explains the variability in the data is a necessary but complex task. This process
involves trying to minimize information loss. From the field of information theory,
several information criteria have been proposed to quantify information, or the
expected value of information, and, among these, the most widely used are the
Akaike information criterion (AIC) (Akaike 1973, 1974) and the Bayesian information criterion (BIC) (Schwarz 1978). Both AIC and BIC are based on the ML
estimates of the model parameters. In a regression fit, the estimates of β´s under
the ordinary least method and the ML method are identical. The difference between
the two methods comes from estimating the common variance σ 2 of the normal
distribution of the errors, around the true mean.
2.5
Specification of a GLM
Table 2.11 Goodness-of-fit
metrics for each of the three
models and regression analysis results for model 3
71
(a) Fit statistics
Model 1
-2 Log likelihood
62.15
AIC (smaller is better)
68.15
AICC (smaller is better)
72.95
BIC (smaller is better)
68.74
CAIC (smaller is better)
71.74
HQIC (smaller is better)
66.87
Pearson’s chi-square
0.50
Pearson’s chi-square / DF
0.07
(b) Type III tests of fixed effects
Model 2
44.49
52.49
62.49
53.27
57.27
50.78
0.07
0.01
Model 3
27.47
37.47
57.47
38.45
43.45
35.34
0.01
0.001
Fvalue
476.73
110.78
50.92
Pr > F
<0.0001
0.0001
0.0008
Effect
Num DF Den DF
Conc
1
5
Conc × conc
1
5
conc × conc × conc 1
5
(c) Parameter estimates
Standard
Effect
Estimate error
DF
Intercept
0.000576 5
0.00040
Conc
0.001946 0.000089 5
conc × conc
2.576E-6 5
0.00003
conc × conc
1.337E-7 2.520e5
× conc
08
Scale
0.001125 0.000530 .
tvalue
0.70
21.83
10.53
5.306
<0.0001
.
.
Pr > |t|
0.5177
<0.0001
0.0001
To get an idea of how to use these adjustment statistics, let us compare three
possible models that best explain the effect of the plasma dilution percentage:
Model 1: ηi = β0 þ β1 × conci
Model 2: ηi = β0 þ β1 × conci þ β2 × conc2i
Model 3: ηi = β0 þ β1 × conci þ β2 × conc2i þ β3 × conc3i
Since the proposed models have a gamma error structure, the commonly used fit
statistic (R2) in a simple linear regression model is not reported. Part of the results of
this analysis is shown below with various metrics as goodness-of-fit measures:
With regard to the values of the goodness-of-fit metrics (Table 2.11 part (a)), the
smaller they are, the better the fit. Based on the above, the accuracy of the fit of the
three regression models increased as the polynomial in the linear predictor increased.
That is, model three best explained the variability of the plasma clotting time. The
72
2
Generalized Linear Models
Fig. 2.12 Fitting the gamma regression model with three predictors
type III sum of squares for fixed effects and the estimated parameters under model
three are tabulated in parts (b) and (c) in Table 2.11, respectively.
Parameter estimates under the linear predictor with linear, quadratic, and cubic
effects are highly significant. The results suggest that a cubic effect for the percentage dilution in plasma concentration in the linear predictor is more efficient in
explaining the clotting time than taking only a linear predictor with only linear or
both linear and quadratic effects (Fig. 2.12).
2.5.5
Beta Regression
Studies in various areas of knowledge, including agriculture, often face the need to
explain a variable expressed as a proportion, percentage, rate, or fraction in the
continuous range (0,1). In economics, for example, the factors that influence the
proportion of households that do not have a cement floor have been studied.
Similarly, in plant breeding, it is desired to investigate the factors that influence
the proportion of plant leaves damaged by a certain disease. In parallel, the proportion of impurities in chemical compounds is of everyday interest in physics and
chemistry. While studies on electoral preferences analyze citizen participation rates
and the variables that can explain them, in the field of education and academic
performance, we try to explain the proportion of success in standardized tests.
Moreover, it is also used to identify the factors associated with the proportion of
credit used by credit card users. The public health field has also been confronted with
the need to model the proportion of coverage in health programs in order to identify
the sociodemographic and economic characteristics associated with whether a
woman is covered. Johnson et al. (1995) presented the properties of the probability
distribution of this type of variable; these researchers showed that the beta distribution can be used to model proportions, since its density can take different forms
depending on the values of the two shape parameters that index the distribution.
However, the beta regression that results from using the beta distribution as a
2.5
Specification of a GLM
73
response variable in the context of generalized linear models is not very well known,
but its use is increasing every day, thanks to friendly software that allow its
implementation in an extremely easy manner.
Definition Let y be a continuous random variable defined in the interval [0, 1] and
α, β > 0. Then, Y has a beta distribution with parameters of forms α and β if and
only if:
f Y ð yÞ =
1
y α - 1 ð 1 - yÞ β - 1 , 0 < y < 1
Bðα, βÞ
ÞΓðβÞ
where B(α, β) is the beta function defined as Bðα, βÞ = ΓΓððααþβ
Þ and Γ is the gamma
function. The mean and variance of this probability density function are given by
E ðY Þ =
α
αβ
:
and VarðY Þ =
αþβ
ðα þ β þ 1Þðα þ βÞ2
In the context of regression analysis, the density of the beta distribution provided
above is not very useful for modeling the mean of the response. Therefore, this
density is reparametrized so that it contains a precision (or dispersion) parameter.
α
and ϕ = α + β, i.e., α = μϕ
This reparameterization consists of defining a μ = αþβ
and β = (1 - μ)ϕ, which means that:
E ðyÞ = μ
and
VarðyÞ =
μ ð1 - μ Þ
1þϕ
So, μ is the mean of the response variable and ϕ can be interpreted as a parameter
of precision in the sense that, for a fixed μ, the higher the value of ϕ, the smaller the
variance of y. The density function of y can be written as:
f ðy; μ, ϕÞ =
ΓðϕÞ
yμϕ - 1 ð1 - yÞð1 - μÞϕ - 1 , 0 < y < 1
ΓðμϕÞΓðð1 - μÞϕÞ
where 0 < μ < 1 and ϕ > 0.
Let y1, y2, . . . , yn be independent and identically distributed random variables,
where each yi with i = 1, 2, . . . , n is modeled under the parametrized beta model
with a mean μ and an unknown parameter ϕ. The model is obtained by assuming that
the mean of yi can be written as:
74
2
Table 2.12 Proportion of
fruit damage ( y) as a function
of concentration. Percentage is
equal to proportion ×100
Generalized Linear Models
Concentration
0.1
0.25
0.5
1
2
4
5
8
10
25
50
100
Y
0.08
0.09
0.11
0.2
0.3
0.53
0.63
0.71
0.73
0.84
0.85
0.86
k
gðμi Þ =
xij βi = ηi
i=1
where β1, β2, . . ., βk are unknown regression parameters and xij are the k covariates
(k < n) that are fixed and known. Finally, g(∙) is a strictly monotone and differentiable link function that maps to the real numbers in the interval (0, 1).
There are several possible options for the link function g(∙). For example, we can
use a logit link function gðμÞ = log
μ
1-μ
, which is considered the most popular and
asymptotically efficient, but it is also feasible to use the probit g(μ) = Φ-1(μ)
function, where Φ(∙) is the cumulative distribution function of a standard normal
random variable, and the complementary link function g(μ) = log {- log (1 - μ)},
among others (McCullagh and Nelder 1989).
Example 1 The objective of this experiment was to evaluate the effect of the
concentration of a chemical compound on the proportion of damage ( y) in the fruits
(Table 2.12). This compound is known to inhibit the growth of an insect, but, at a
certain concentration, it can cause damage to the fruits.
The proportion of damage to the fruits can be modeled under a beta distribution
(μ, ϕ). Let yi be the proportion of damage to the fruits due to the ith concentration.
The GLM components for this dataset are as follows:
Distribution: yi betaðμi , ϕÞ, with E ðyÞ = μ
and VarðyÞ =
μð1 - μÞ
1þϕ
Linear predictor: ηi = β0 þ β1 × conci
Link function : ηi = logitðπ i Þ = log
πi
ðlogit logÞ
ð1 - π i Þ
2.5
Specification of a GLM
Table 2.13 Goodness-of-fit
metrics for the linear and quadratic models and results of
the quadratic model fit
75
(a) Fit statistics
Linear
Quadratic
-2 Log likelihood
-6.23
-14.50
AIC (smaller is better)
-0.23
-6.50
AICC (smaller is better)
2.77
-0.79
BIC (smaller is better)
1.22
-4.56
CAIC (smaller is better)
4.22
-0.56
HQIC (smaller is better)
-0.77
-7.22
Pearson’s chi-square
12.85
15.05
Pearson’s chi-square / DF
1.07
1.25
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Conc
1
8.08
0.0193
Conc × conc
1
6.11
0.0354
(c) Parameter estimates
Standard
terror
Effect
Estimate
DF value Pr > |t|
Intercept
-1.1425
0.2935
-3.89 0.0037
Conc
0.1572
0.05530
2.84 0.0193
Conc ×
-0.00132 0.000534
-2.47 0.0354
conc
Scale
9.0432
4.0045
.
.
.
Note that we are using conc to denote the independent variable concentration of
the chemical compound. The following SAS code allows us to perform a beta
regression for the dataset:
proc glimmix method=laplace;
model y = conc / dist=beta s;
run;
The “method = Laplace” statement asks SAS for the estimation method to be
Laplace integration, and the “dist = beta” and “s” options invoke GLIMMIX to
perform beta regression and provide fixed parameter estimation, respectively.
In order to see which type of linear, quadratic, or cubic predictor best explains the
observed variability in a dataset, we make use of the fit statistics (-2 log likelihood,
AIC, etc.). Part of the output is shown below in Table 2.13. According to the fit
statistics in part (a), the predictor that best models this experiment is the quadratic
predictor.
In Fig. 2.13, we can see that the best linear predictor to model a dataset is of the
cubic order, but due to the indeterminacy (not showing here) in the t-value (infinity),
in the hypothesis test of the estimated parameters, it was decided to take the
quadratic predictor. Both predictors, quadratic and cubic, better model the proportion (percentage = proportion×100) of fruit damage caused by the concentration of
the applied chemical.
76
2
Generalized Linear Models
Fig. 2.13 Fitting the beta regression model
2.6
Exercises
Exercise 2.6.1 The partial dataset corresponds to an evaluation of the effects of
increasing application rates of picloram (0, 1.1, 2.2, and 4.5 kg/ha) for the control of
larkspur plants (data in Table 2.14). The objective of this study was to study the
efficacy of picloram herbicide in controlling larkspur plants.
(a) List and describe the components of the GLM (distribution, systematic component (predictor), and the link function).
(b) Fit the model according to part (a).
(c) Interpret your results.
Exercise 2.6.2 Effect of pH, Brix, temperature, and nisin concentration on the
growth of Alicyclobacillus acidoterrestris CRA7152 in apple juice. The objective
of this experiment was to model the presence/absence of CRA7152 growth in apple
juice as a function of pH (3.5–5.5), Brix (11–19), temperature (25–50 °C), and nisin
concentration (0–70). The data are shown below (Table 2.15):
(a) List and describe the components of the GLM (distribution, systematic component (predictor), and the link function).
(b) Fit the model according to part (a).
(c) Interpret your findings.
Exercise 2.6.3 The objective of this experiment was to evaluate the level of toxicity
of concentrations of pyrethrin and piperonyl butoxide on the mortality of beetles
(Tribolium castaneum). Pyrethrin is a natural insecticide found in the plant Chrysanthemum cinerariaefolium and its flowers. The active ingredients are pyrethrins I
and II, cinerins I and II, and jasmolins I and II. The dried flowers contain 0.9–1.3%
pyrethrum. The crude extract contains 50–60% pyrethrum and is imported from
2.6
Exercises
77
Table 2.14 Toxicity of picloram in controlling larkspur plants
Rep
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Conc
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
Rep
Conc
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
Rep
Conc
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(continued)
78
2
Generalized Linear Models
Table 2.14 (continued)
Rep
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Conc
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
Y
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Rep
Conc
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
Y
1
1
1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Rep
Conc
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
Y
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
(continued)
2.6
Exercises
79
Table 2.14 (continued)
Rep
1
1
1
1
1
1
1
1
1
1
1
Conc
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
Y
1
1
1
1
1
1
1
1
1
1
1
Rep
Conc
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
Y
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Rep
Conc
2.2
2.2
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
Y
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Rep replicate, Conc concentration, Y =1 dead/ Y = 0 alive
various countries. The extract is diluted to 20%, which is the maximum concentration commercially available in the United States. Pyrethrin oxidizes on exposure to
air but has been shown to be stable for long periods in water-based emulsions and oil
concentrates. Synergistic compounds (such as piperonyl butoxide or N-octyl
bicycloheptene dicarboximide), which enhance the effect of pyrethrin on insects,
are present in commercially available pyrethrin formulations. The results of this
study are shown below (Table 2.16).
(a) List and describe the components of the GLM (distribution, systematic component (predictor), and the link function).
(b) Fit the model according to part (a).
(c) Interpret your results.
80
2
Generalized Linear Models
Table 2.15 Growth of Alicyclobacillus acidoterrestris CRA7152
pH
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
3.5
3.5
3.5
3.5
3.5
3.5
3.5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
3.5
4
5
Nisin
70
70
50
50
30
30
0
0
70
70
50
50
30
30
0
70
70
50
50
30
30
0
0
70
70
50
50
30
30
30
0
0
0
0
0
Temp (°C)
50
43
43
35
35
25
50
25
43
35
50
35
50
43
25
25
25
50
25
43
43
50
35
50
35
43
25
50
35
25
43
43
35
35
43
Brix
11
19
13
15
13
11
19
15
15
11
13
19
11
15
19
15
13
15
19
19
11
13
11
19
13
11
11
15
19
13
15
13
11
11
11
Y
0
0
1
1
1
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
1
1
1
0
0
0
0
1
0
0
1
1
0
1
1
pH
5.5
3.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
3.5
3.5
3.5
3.5
3.5
3.5
3.5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
3.5
4
5
5.5
3.5
Nisin
70
0
70
70
50
50
30
30
0
0
70
70
50
50
30
30
0
70
70
50
50
30
30
0
0
70
70
50
50
30
30
30
0
0
0
0
0
70
0
Temp (°C)
50
25
50
43
43
35
35
25
50
25
43
35
50
35
50
43
25
25
25
50
25
43
43
50
35
50
35
43
25
50
35
25
43
43
35
35
43
50
25
Brix
19
11
11
19
13
15
13
11
19
15
15
11
13
19
11
15
19
15
13
15
19
19
11
13
11
19
13
11
11
15
19
13
15
13
11
11
11
19
11
Y
0
0
0
0
1
1
1
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
1
1
1
0
0
0
0
1
0
0
1
1
0
1
1
0
0
2.6
Exercises
Table 2.16 Mixture: pyrethrin plus piperonyl; n is the
number of beetles exposed
and Y is number of beetles
killed
81
n
150
149
150
151
151
150
149
150
140
150
150
149
200
Mixture
1.5
1.06
0.75
1.35
1.03
0.8
3.3
3.07
2.9
10.65
10.46
10.32
0
Y
138
75
32
129
65
19
143
112
37
141
117
56
1
Table 2.17 Results of the experiment with carbon disulfide
Dose
49.1
53
56.9
60.8
64.8
68.7
72.6
76.5
Number of exposed beetles
59
60
62
56
63
59
62
60
Number of dead beetles
6
13
18
28
52
53
61
61
Proportion of dead beetles
0.102
0.217
0.29
0.5
0.825
0.898
0.984
1
Exercise 2.6.4 The objective of this experiment was to model the probability of
mortality of the toxic effect of carbon disulfide (CS2) gas on beetles. The insects
were exposed to various concentrations of this gas (in mf/L) for 5 hours (Bliss 1935),
and, then the number of dead beetles (Y ) was counted. The data are shown below
(Table 2.17).
(a) List and describe the components of the GLM (distribution, systematic component (predictor), and the link function).
(b) Fit the model according to part (a).
(c) Interpret your results.
Exercise 2.6.5 A study was conducted to assess the fowlpox virus in chorioallantois
by the Pock counting technique. The membrane Pock count for 50 embryos exposed
to one of four dilutions of virus (multiples of 10ˆ(-3.86)). The FD column heading
corresponds to the dilution factor and the number of Pocks observed (Table 2.18).
82
2
Generalized Linear Models
Table 2.18 Results of the fowl pox experiment
FD
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
Count
1
2
2
3
2
2
1
0
0
1
2
1
1
2
1
Table 2.19 Number of
reversed Salmonella TA98
colonies
FD
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
Count
5
2
3
2
5
0
2
2
0
3
2
2
FD
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
Quinoline dosage (μg/placa)
0
10
33
15
16
16
21
18
26
19
21
33
Count
5
11
7
5
4
6
5
9
4
7
4
8
4
FD
1
1
1
1
1
1
1
1
1
1
Count
12
9
11
17
11
10
8
16
15
12
100
27
41
60
333
33
38
41
1000
20
27
42
(a) List and describe the components of the GLM (distribution, systematic component (predictor), and the link function).
(b) Fit the model according to part (a).
(c) Interpret your findings.
Exercise 2.6.6 Data were provided by Margolin et al. (1981) from an Ames
Salmonella reverse mutagenicity assay. The table shows the number of reversed
colonies observed on each of the three plates (repeats) tested at each of the six
quinoline dose levels. The focus is on testing for mutagenic effects over time in the
excess variation typically observed between counts (Table 2.19).
(a) List and describe the components of the GLM (distribution, systematic component (predictor), and the link function).
(b) Fit the model according to part a).
(c) Interpret your results.
Appendix
83
Appendix
Students infected with a certain disease
id
Day
Students
id
1
1
6
45
2
2
8
46
3
3
12
47
4
3
9
48
5
4
3
49
6
4
3
50
7
4
11
51
8
6
5
52
9
7
7
53
10
8
3
54
11
8
8
55
12
8
4
56
13
8
6
57
14
12
8
58
15
14
3
59
16
15
6
60
17
17
3
61
18
17
2
62
19
17
2
63
20
18
6
64
21
19
3
65
22
19
7
66
23
20
7
67
24
23
2
68
25
23
2
69
26
23
8
70
27
24
3
71
28
24
6
72
29
25
5
73
30
26
7
74
31
27
6
75
32
28
4
76
29
4
77
33
34
34
3
78
35
36
3
79
36
36
5
80
37
42
3
81
38
42
3
82
39
43
3
83
40
43
5
84
Day
45
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88
88
90
93
Students
3
3
3
2
3
1
3
3
5
4
4
3
5
4
3
5
3
4
2
3
3
1
3
2
5
4
3
0
3
3
4
0
3
3
4
0
2
2
1
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id
89
90
91
92
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95
96
97
98
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100
101
102
103
104
105
106
107
108
109
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Students
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0
0
1
1
2
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1
1
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0
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0
1
1
0
0
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0
0
(continued)
84
Students infected with a certain disease
id
Day
Students
id
41
44
3
85
42
44
5
86
43
44
6
87
44
44
3
88
2
Day
93
94
95
95
Students
2
0
2
1
Generalized Linear Models
id
Day
Students
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the copyright holder.
Chapter 3
Objectives of Inference for Stochastic
Models
Throughout this book, we have been using the pseudonym GLMMs to denote
generalized linear mixed models. The common denominator among all these models
is that they all contain a linear model (LM) part, which refers to the fixed effects
component of the linear predictor Xβ. In a GLMM, the prefix “G” indicates that the
distribution of observations may not be normal, the suffix of the first M means that
the linear predictor includes mixed effects and thus contains random effects, which
are expressed by the term “Zb.” The fixed linear component of the predictor Xβ is
important because the fixed effects describe the treatment design, which, in turn, is
determined by the objectives or the initial research questions that the study wishes to
answer. Therefore, if the researcher proposes using a reasonable model to analyze an
experiment, then he/she must be able to express each objective as a question about a
model parameter or as a linear combination of model parameters.
Example Assume a factorial 2 × 2 model, with two levels in both factors A and B,
in which all possible combinations are tested. In this case, Xβ corresponds to a
two-way model with interaction and a predictor given by
ηij = μ þ αi þ βj þ ðαβÞij ; i, j = 1, 2
As in all the statistical models studied so far, the linear predictor is expressed in
terms of the link function, and ηij can estimate the mean μij (a combination of
treatments) directly if the data follow a normal distribution and indirectly if the
data are not normally distributed. For this example, the inference should focus on
one or more of the following options (estimable functions): a treatment combination
mean; a main effect mean; the mean of factor A, which is the average of the overall
levels of factor B or vice versa; the difference of the main effects or the difference of
a single effect, i.e., the difference between two levels of factor A at a given level of B
or the difference between two levels of factor B at a given level of factor A; and so
on. Each of these options can be expressed in terms of the parameters of the linear
predictor, as shown in Table 3.1.
© The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8_3
85
86
3
Objectives of Inference for Stochastic Models
Table 3.1 Estimable functions in a factorial 2 × 2 treatment structure using the identity link
function
Target estimate
Combination A × B
Main effect of factor A
Parameter estimator of the linear predictor
η + αi + βj + (αβ)ij
ηi: = η þ αi þ 12 βj þ 12 ðαβÞij
Target estimation in
terms of the expected
value
μij
μ
i: = 12 μij
Main effect of factor B
η:j = η þ 12
μ
:j =
j
Difference between
level 1 and level 2 of
factor A
Difference between
level 1 and level 2 of
factor B
Simple effect of A
given Bj (A| Bj)
Simple effect of B
given A (B| Ai)
Interaction between
factors A and B
(A × B) = A|B1 - A|B2
=B j A1 - B j A2
i
αi þ βj þ 12
η1: - η2: = α1 - α2 þ 12
η:1 - η:2 =
1
2
i
ðαβÞi1 -
j
ðαβÞij
i
j
ðαβÞ1j -
i
j
ðαβÞ2j
ðαβÞi2 þ β1 - β2
j
1
2
μij
i
μ
1: - μ
2:
μ
:1 - μ
:2
η1j - η2j = α1 - α2 + (αβ)1j - (αβ)2j
μ1j - μ2j
ηi1 - ηi2 = (αβ)i1 - (αβ)i2 + β1 - β2
μi1 - μi2
(η11 - η21) - (η12 - η22) = (αβ)11 - (αβ)21
- (αβ)12 + (αβ)22= (η11 - η12) - (η21 - η22)
μ11 - μ12 - μ21 + μ22
Assuming that the data have a normal distribution, which is equivalent to using an
identity link function, the estimator, in terms of the linear predictor (column 2),
estimates the expected values of column three. If the data do not follow a normal
distribution, then column 2 indirectly estimates the expected values of column three,
and, in order to estimate the expected values, link functions are required. For link
functions other than identity, the estimates in column two require a more careful
handling. In an experimental design with a factorial treatment structure, the analysis
should focus on the interaction of the two factors. If this interaction is significant,
then the simple effects are not equal; however, if the interaction is not significant,
then the main effects provide useful information; otherwise, the main effects are
confounded. Therefore, for this reason, in this case, it is better to focus on the simple
effects.
3.1
Three Aspects to Consider for an Inference
When constructing a model, the researcher must decide whether the effects are fixed
or random. This decision has important implications with respect to the estimation
criteria and in the interpretation of the tests and estimates obtained. Given these
implications, three important aspects, described in the following sections, must be
taken into consideration in statistical modeling.
3.1
Three Aspects to Consider for an Inference
3.1.1
87
Data Scale in the Modeling Process Versus
Original Data
This is a very particular issue for models with a link function other than “identity,”
since the scale of the data used in the modeling process is not always the same as the
scale of the original data when the assumption of normality in the response variable
is no longer valid. When the data are normally distributed, the estimable function
directly estimates the expected value. However, this is not true if the data follow a
non-normal distribution. For example, in a logistic model for binomial data in a
completely randomized design, the estimable function η + τi estimates a logit or
“log” odds. In this vein, η + τi must be expressed as a probability and not as a logit,
i.e., the expected value for individuals receiving the ith treatment is a probability.
This requires converting the estimate to a probability, using the inverse link; that is:
πi =
1
ð1 þ e - ðηþτi Þ Þ
Thus, for functions other than “identity,” there are two ways of expressing the
estimates: (1) in terms of the parameters directly estimated from the GLMM (model
scale) or (2) in terms of the expected value of the response variable (data scale).
3.1.2
Inference Space
This problem arises only when the linear predictor contains random effects. In these
models, the estimates are obtained through a linear combination (an estimable
function) with fixed effects, even though the linear predictor contains random
effects. K′β denotes the estimable function, where K is the matrix of order
[( p + 1) × k] and β is the vector of fixed effects parameters of order [( p + 1) × 1].
The estimable function (K′β) represents a broad inference as it generalizes results to
the entire population represented by the random effects.
Although the linear combination K′β + Z′b is a predictable function with Z′, a
matrix for random effects with nonzero coefficients, its inference is limited to only
those levels defined in b. Suppose that you are conducting an experiment with three
treatments at different locations (L ), then the estimable function τ1 - τ2 provides
information for the inference about the difference between treatments 1 and 2 in the
whole population under study. Although the predictable function [τ1 - τ2 + (Lτ)1j (Lτ)2j] constrains the inference space between treatments 1 and 2, it is limited to
location (Lj). The type of inference produced by predictable functions is called
“narrow inference” because the nonzero coefficients in matrix Z reduce the scope
of inference for the entire population at those levels identified in Z. Thus, the
predictive function K′β + Z′b should be used for specific estimates, whereas the
estimable function K′β should be used for valid estimates for the entire population
under study.
88
3.1.3
3
Objectives of Inference for Stochastic Models
Inference Based on Marginal and Conditional Models
As mentioned in the previous chapter, the specification of a generalized linear mixed
model (GLMM) is done in terms of two probability distributions: (1) the distribution
of the observations, given the random effects y j b and (2) the distribution of the
random effects b. This feature is very particular to Gaussian (and non-Gaussian)
mixed models (MMs), for this reason, it is also valid for mixed models with response
variables that are different from a normal distribution.
From the probability theory, the marginal probability distribution of data (y) can
be obtained by integrating over the random effects, b, from the joint probability
distribution of y and b. Of the two distributions, the marginal distribution of data is
the only one that can be known and observable. Many non-Gaussian mixed models,
which seem reasonable, do not distinguish between the distribution of y j b and y.
Models that do not make this distinction are called marginal models. Estimates
obtained by marginal models have different expected values compared to those
produced by conditional models. Therefore, marginal models are not estimated in
the same way as conditional models.
3.2
Illustrative Examples of the Data Scale
and the Model Scale
In linear models, inference begins with the estimable function K′β, and, these
models, in turn, are defined in terms of the linear function η = g(μ) = Xβ (if there
are no random effects) and η = Xβ + Zb (if there are random effects in the model),
whereas K′β produces results in terms of the link function.
For linear normal response models such as LMs and LMMs, the link function is
not visible because they use the “identity” function as the link. Linear combinations
of model parameters directly estimate desired values such as differences between
treatments and many other hypothesis tests of interest. Inference for an LM is
straightforward.
For GLMs and GLMMs with a non-normal response, the estimation of K′β yields
a linear combination of elements of the linear predictor η, which is a linear combination of g(μ), typically a nonlinear function of μ. For example, with Poisson data the
K′β is a function of logarithm (log) and for binomial data, it is a function of logit or
probit. However, most of the time, the researcher wants to see the binomial results
expressed in terms of the probability of the outcome of interest, whereas for Poisson,
the results are expressed in terms of counts. This means that since both GLMs and
GLMMs carry out the estimation process on the scale of the model (depending on the
link used) to report the results of interest in terms of the scale of the data, it is
necessary to apply the inverse link to the predictor in terms of the model scale to
express the results. To mention two examples, in the case of the logit link for
binomial, the results are expressed in terms of probability and, in the case of the
3.2
Illustrative Examples of the Data Scale and the Model Scale
89
Table 3.2 Percentage of germinated seeds (Y ) out of total seeds (N )
Treatment (Trt)
Trt1
Trt1
Trt1
Trt2
Trt2
Trt2
Trt3
Trt3
Trt3
Y (no. of germinated seeds)
54
41
52
28
22
21
41
37
47
N (total no. of seeds)
70
60
70
70
60
70
70
60
70
Poisson model, they are expressed in terms of counts. To exemplify the model scale
and the data scale, an example is shown below.
Example 3.1 Consider the following experiment in which three chemical seed coat
softeners were tested for studying their effect on germination of tomato seeds in
Styrofoam trays (Table 3.2).
To illustrate the above two concepts, we first analyze these data using a
completely randomized design (CRD), assuming the response variable to be normal,
and, then, we analyze the same experimental design but with a binomial response
variable. We are interested in comparing the means of treatments using a completely
randomized design. Note that for demonstrative purposes, we are assuming that
Y has a normal distribution, when in fact it has a binomial distribution.
The components of this model are defined as follows:
Distribution: yij~N(μi, σ 2)
Linear predictor: ηi = η + τi; (i = 1, 2, 3)
Link function: ηi = μi (identity link)
The analysis of variance (ANOVA) (part (a)) and estimated parameters (part (b))
of this experimental design indicate that there is a highly significant difference
between the treatments (P = 0.0033) for the germination of tomato seeds.
Table 3.3 shows part of the results.
The estimated parameter values of the model, except for treatment three, are
shown in the table above (obtained with the “solution” command) because the model
is over-parameterized. The estimable functions K′β for the treatment means are as
follows:
K0 =
1
1
1
1
0
0
0 0
1 0
0 1
; β=
η
τ1
τ2
τ3
90
3
Objectives of Inference for Stochastic Models
Table 3.3 Results of the analysis of variance using a CRD
(a) Type III tests of fixed effects
Effect
Num degree of freedom (DF)
Trt
2
(b) Parameter estimates
Effect
Trt
Estimate
^η
Intercept
41.6667
7.3333
Trt
Trt1
^τ1
Trt
Trt2
^τ2
-18.0000
0.
Trt
Trt3
^τ3
Scale
29.5556
Den DF
6
Standard error
3.1388
4.4389
4.4389
..
17.0639.
F-value
17.25
DF
6
6
6
.
.
t-value
13.27
1.65
-4.06
Pr > F
0.0033
Pr > |t|
<0.0001
0.1496
0.0067
.
From the estimated treatment parameters τi = μ
i = ^η þ ^τi , we can obtain the
estimated mean for each one of the treatments (i = 1, 2, 3) as follows: for treatment
1, τ1 = ^η þ ^τ1 = 41:6667 þ 7:3333 = 49; for treatment
2, τ2 = ^η þ ^τ2 = 41:6667 - 18 = 23:6667; and for treatment
3, τ3 = ^η þ ^τ3 = 41:6667 þ 0 = 41. The value of the mean squared error (^
σ 2 ),
which appears in the table as “Scale,” is 29.5556.
For the difference between treatments, the τi - τi′ values for i ≠ i′are as follows:
τ1 - τ2 = ^η þ ^τ1 - ð^η þ ^τ2 Þ = ^τ1 - ^τ2 = 7:3333 - ð- 18Þ = 25:3333,τ1 - τ3 = ^η þ ^τ1
- ð^η þ ^τ3 Þ = ^τ1 - ^τ3 = 7:333 - 0:0 = 7:3333, and τ2 - τ3 = ^η þ ^τ2 - ð^η þ ^τ3 Þ = ^τ2
- ^τ3 = - 18:00 - 0:0 = - 18:0. These estimates can be obtained using the Statistical Analysis Software (SAS) “estimate” and “lsmeans” commands, as shown
below:
proc glimmix data=germi;
class trt;
model y=trt / solution;
lsmeans trt / diff e;
estimate 'lsm trt 1' intercept 1 trt 1 0;
estimate 'lsm trt 2' intercept 1 trt 0 1;
estimate 'lsm trt 3' intercept 1 trt 0 0 1;
estimate 'overall mean' intercept 1 trt 0.33333 0.33333 0.33333
0.33333;
estimate 'overall mean' intercept 3 trt 1 1 1 1 / divisor=3;
estimate 'trt diff 1&2' trt 1 -1 0;
estimate 'trt diff 2&3' trt 0 1 -1;
run;
The “estimate” command requires us to specify what we wish to estimate and the
“intercept” command refers to the intercept (η) and “Trt” to the treatment (τi) effects
under evaluation; the coefficients needed for the estimates are shown above. While
the “lsmeans” command invokes GLIMMIX in SAS to estimate the treatment
means, “diff” asks to estimate the differences between pairs of treatments, and “E”
3.2
Illustrative Examples of the Data Scale and the Model Scale
91
Table 3.4 Results obtained using the “estimate” and “lsmeans” commands
(a) Differences of Trt least squares means
Trt
Trt
Estimate
Standard error
Trt1
Trt2
25.3333
4.4389
Trt1
Trt3
7.3333
4.4389
Trt2
Trt3
-18.0000
4.4389
(b) Estimates
Label
Estimate
Standard error
LSM Trt 1
49.0000
3.1388
LSM Trt 2
23.6667
3.1388
LSM Trt 3
41.6667
3.1388
Overall mean
38.1111
1.8122
Overall mean
38.1111
1.8122
Trt diff 1&2
25.3333
4.4389
Trt diff 2&3
-18.0000
4.4389
DF
6
6
6
DF
6
6
6
6
6
6
6
t-value
5.71
1.65
-4.06
t-value
15.61
7.54
13.27
21.03
21.03
5.71
-4.06
Pr > |t|
0.0013
0.1496
0.0067
Pr > |t|
<0.0001
0.0003
<0.0001
<0.0001
<0.0001
0.0013
0.0067
displays the coefficients of the estimable functions used in “lsmeans.” Some of the
outputs of the above code are shown in Table 3.4.
Next, we analyze the same data, also using a CRD, but now assuming a binomial
distribution in the response variable. N indicates the independent number of
Bernoulli trials observed in the ijth observation. The components of the model are
as follows:
Distribution: yij~Binomial(Nij, π i)
Linear predictor: ηi = η + τi; (i = 1, 2, 3)
Link function: ηi = logit
πi
1 - πi
(logit link)
Fitting these data in a binomial model, the fixed effects solution of the parameters
obtained in terms of the model scale are tabulated in Table 3.5.
The above results were obtained using the following SAS code:
proc glimmix data=germi;
class trt;
model y/n=trt / solution;
run;
Similar to the previous example, we can estimate the mean of treatments and the
differences between two pairs of treatments. The linear predictors for the treatments
are as follows: ^η1 = ^η þ ^τ1 = 0:5108 þ 0:5093 = 1:0201, ^η2 = ^η þ ^τ2 = 0:5108 1:108 = - 0:5971, and ^η3 = ^η þ ^τ3 = 0:5108 þ 0:0 = 0:5108, and, for the differences between treatments (1 and 2, 1 and 3, and 2 and 3), they are as follows:
^η1 - ^η2 = 1:0201 - ð- 0:5971Þ = 1:6173, ^η1 - ^η3 = 1:0201 - 0:5108 = 0:5093, and
^η2 - ^η3 = - 0:5971 - 0:5108 = - 1:1079, respectively
92
3
Objectives of Inference for Stochastic Models
Table 3.5 Estimated parameters at the model scale
Effect
Intercept
Trt
Trt
Trt
Parameter estimates
Estimate
^η
0.5108
0.5093
^τ2
-1.1080
^τ2
0
^τ3
Trt
Trt1
Trt2
Trt3
Standard error
0.1461
0.2168
0.2078
.
DF
6
6
6
.
t-value
3.50
2.35
-5.33
.
Pr > |t|
0.0129
0.0571
0.0018
.
Using the relationship between the linear predictor and the link function ηi =
logit
πi
1 - πi
= log
πi
1 - πi
, we can estimate the probability of observing a favorable
outcome for each of the treatments, that is, π 1, π 2, and π 3, respectively. Applying the
inverse link, we obtain:
π^1 = 1= 1 þ e - ð^ηþ^τ1 Þ ; π^2 = 1= 1 þ e - ð^ηþ^τ2 Þ ; and π^3 = 1= 1 þ e - ð^ηþ^τ3 Þ
Substituting the corresponding values, we obtain
π^1 = 1= 1 þ e - ð^ηþ^τ1 Þ = 1= 1 þ e - 1:0201 = 0:735,
π^2 = 1= 1 þ e - ð^ηþ^τ2 Þ = 1= 1 þ e - ð- 0:5971Þ = 0:355, and
π^3 = 1= 1 þ e - ð^ηþ^τ3 Þ = 1= 1 þ e - ð0:5108Þ = 0:625
Here, we can see that the treatment with the highest probability of success is
treatment one, followed by treatment three, whereas treatment two has the lowest
probability of success. Now, for the difference between two treatments, τi - τi′ for
i ≠ i′, we can estimate the logarithm of the odds ratio as
τi - τi0 = log
πi
π i0
- log
= log
1 - πi
1 - π i0
where, in this particular case, odds =
oddsratio = log
πi
1 - πi
=
π i0
1-π 0
i
πi
1 - πi
πi
1 - πi
=
π0
i
1 - π i0
is the odds of the treatment i and
is the odds ratio for treatments i and i′, for i ≠ i′.
When applying the inverse link to the above expression (odds ratio), we get
Oddsratio = 1=
1þe
ð
- ^τi - ^τ 0
i
Þ
3.2
Illustrative Examples of the Data Scale and the Model Scale
93
The value of the odds ratios for treatments 1 and 3 is
Oddsratio1 - 3 = 1=
1þe - ð^τ1 - ^τ3 Þ
= 1 ð1þe - 0:5093 Þ = 0:6246
Similarly, for the pair of treatments 1–2 and 2–3, the resulting odds ratios are
Oddsratio1 - 2 = 0.8344 and Oddsratio2 - 3 = 0.2483, respectively. It is important to
mention that the odds ratios are not the mean of the difference of π i - π i′for i ≠ i′.
From the previous example, it is clear that when the response variable is not
normal, parameter estimation and inference occurs at two levels. The linear predictor
Xβ and the estimable function K′β are expressed in terms of the link function, logit –
estimates on the model scale (scale of the link function) – as in the above example.
Under the logit link, the logarithm of the odds and the difference of the estimate (log
odds ratio) are very common and useful terms in categorical data analysis for the
estimation of treatments or treatment differences in terms of the data scale.
Commonly, estimation at the model scale in GLMs is not very easy to interpret,
and, as such, the data scale plays a very important role. A data scale involves
applying the inverse of the link function to the estimable function, K′β, as we did
in the previous example to convert the log of the odds for each treatment to a
probability. In general, we use the inverse of the link function to transform the
estimates at the model scale to the data scale. The inverse of the link function is not
used for estimating the differences between treatments because the link functions are
generally nonlinear. This is why the inverse of the link function is not applied to the
differences between treatments because it produces meaningless results.
Thus, in the logit model, we have two approximations for the difference. First, we
could apply the inverse of the link function to each linear predictor for each treatment
and then take the difference between probabilities: π^i - π^i ′ . That is, we can estimate
the difference between π i - π i′ through 1= 1 þ e - ðηþτi Þ - 1= 1 þ e - ðηþτi0 Þ
and not as 1= 1 þ e - ð^τi - ^τi0 Þ . Second, we know that τi - τi′ estimates the
logarithm of the odds ratio by means of eðτi - τi0 Þ , which produces an estimate of
the odds ratio. Both approaches are valid, and the use of one approach or the other
depends on the requirements of the particular study.
With the GLIMMIX procedure, we can implement the solution in terms of the
data scale with the “ilink,” “exp,” and “oddsratio” commands, as shown in the
following SAS code:
proc glimmix;
class trt;
model y/n=trt / solution oddsratio;
lsmeans trt / diff oddsratio ilink ;
estimate 'lsm trt 1' intercept 1 trt 1 0/ilink;
estimate 'lsm trt 2' intercept 1 trt 0 1/ilink;
estimate 'lsm trt 3' intercept 1 trt 0 0 1/ilink;
estimate 'overall mean' intercept 1 trt 0.33333 0.33333 0.33333
0.33333/ilink;
estimate 'overall mean' intercept 3 trt 1 1 1 1 / divisor=3 ilink;
94
3
Objectives of Inference for Stochastic Models
estimate 'trt diff 1&2' trt 1 -1 0/ oddsratio ilink;
estimate 'trt diff 1&3' trt 1 0 -1/oddsratio ilink;
estimate 'trt diff 2&3' trt 0 1 -1/oddsratio ilink;
estimate 'trt diff 1&2' trt 1 -1 0/exp;
estimate 'trt diff 1&3' trt 1 0 -1/exp;
estimate 'trt diff 2&3' trt 0 1 -1/exp;
run;
Part of the output of “proc GLIMMIX” is shown in Table 3.6. The “Odds ratio
estimates” (part (a)) are the result of the “oddsratio” command in the previous
program, whereas the confidence intervals are provided by default.
What appears under “Estimate” (in part (b)) is in the model scale ^ηi = ^η þ ^τi , and
what appears under “Mean” (in part (b)) is an estimate of the inverse of the link
function π^i = 1= 1 þ e - ð^ηþ^τi Þ and, in this case, is a probability that corresponds to
the data scale. Similarly, what appears under “Estimate” is in model scale ^τi - ^τi0 ,
whereas the “Odds ratio” values were estimated using eðτi - τi0 Þ and are in data scale.
Under “Estimates” column in Table 3.7, the log odds ratio appears as an
“Exponentiated estimate” regardless of whether we use the “oddsratio” or “exp”
option in the “estimate” command. For the overall mean, the inverse of the link
function applied to ^η þ 13 ð^τ1 þ ^τ2 þ ^τ3 Þ is 0.5772, which is totally different from the
average of π^is; that is, 13 ðπ^1 þ π^2 þ π^3 Þ = 13 ð0:735 þ 0:355 þ 0:655Þ = 0:5816. This
illustrates that we have to be extremely careful when using the output of proc
GLIMMIX, as it can produce outputs in terms of both the model scale and the
data scale through the application of the inverse of the link function; however, this
has to be applied appropriately, otherwise, we will get meaningless results.
Example 3.2: Randomized complete block design (RCBD) with normal
and binomial responses
Now, assume that we have the same example but in an RCBD. The three treatments
were tested in each of the blocks, as shown in Table 3.8.
In this example, first, the data are analyzed assuming a normal response and
assuming that the block effect is fixed; then, they are analyzed assuming a binomial
response.
The model components under a Gaussian response variable are as follows:
Distribution: yij ~ N(μij, σ 2)
Linear predictor: ηij = η + τi + blockj; (i, j = 1, 2, 3)
Link function: ηij = μij; (identity link)
From the theory of linear models, we know that we can estimate the ith treatment
mean through
3
ηi • = 1=3
3
yij = η þ τi þ 1=3
j=1
j=1
blockj = η þ τi þ bloq •
3.2
Illustrative Examples of the Data Scale and the Model Scale
95
Table 3.6 Results of the “ilink,” “exp,” and “oddsratio” commands
(a) Odds ratio estimates
Trt
Trt
Estimate
DF
95% confidence limits
Trt1
Trt3
1.664
6
0.979
2.829
Trt2
Trt3
0.330
6
0.199
0.549
(b) Trt least squares means
Trt
Estimate Standard error DF t-value Pr > |t| Mean
Standard error Mean
Trt1
1.0201 0.1602
6
6.37
0.0007
0.7350 0.03121
Trt2 -0.5971 0.1478
6
-4.04
0.0068
0.3550 0.03384
Trt3
0.5108 0.1461
6
3.50
0.0129
0.6250 0.03423
(c) Differences of Trt least squares means
Trt
Trt
Estimate
Standard error
DF
t-value
Pr > |t|
Odds ratio
Trt1
Trt2
1.6173
0.2180
6
7.42
0.0003
5.039
Trt1
Trt3
0.5093
0.2168
6
2.35
0.0571
1.664
Trt2
Trt3
-1.1080
0.2078
6
-5.33
0.0018
0.330
Table 3.7 Estimates at the model scale and at the data scale
Estimates
Label
LSM
Trt 1
LSM
Trt 2
LSM
Trt 3
Overall
Mean
Trt diff
1&2
Trt diff
1&3
Trt diff
2&3
Trt diff
1&2
Trt diff
1&3
Trt diff
2&3
Estimate
1.0201
Standard
error
0.1602
DF
6
t Value
6.37
Pr > |t|
0.0007
Mean
0.7350
Standard
error
Mean
0.03121
-0.5971
0.1478
6
-4.04
0.0068
0.3550
0.03384
0.5108
0.1461
6
3.50
0.0129
0.6250
0.03423
0.3113
0.08746
6
3.56
0.0119
0.5772
0.02134
1.6173
0.2180
6
7.42
0.0003
0.8344
0.03011
5.0393
0.5093
0.2168
6
2.35
0.0571
0.6246
0.05083
1.6642
-1.1080
0.2078
6
-5.33
0.0018
0.2483
0.03878
0.3302
1.6173
0.2180
6
7.42
0.0003
5.0393
0.5093
0.2168
6
2.35
0.0571
1.6642
-1.1080
0.2078
6
-5.33
0.0018
0.3302
Exponentiated
estimate
96
3
Objectives of Inference for Stochastic Models
Table 3.8 Percentage of germinated seeds (Y ) out of total seeds (N ) in a randomized complete
block design
Treatment
Trt1
Trt1
Trt1
Trt2
Trt2
Trt2
Trt3
Trt3
Trt3
Block
Block1
Block2
Block3
Block1
Block2
Block3
Block1
Block2
Block3
where bloq • = 1=3
3
j=1
Y (no. of germinated seeds)
54
41
52
28
22
21
41
37
47
N (total no. of seeds)
70
60
70
70
60
70
70
60
70
blockj :
For the mean difference of two treatments i and i’, this is estimated as
• - ðη þ τi0 þ bloq
• Þ = τi - τi ′
ηi • - ηi ′ • = η þ τi þ bloq
The goal of this experiment could be to compare the treatment means, that is,
η1: = η2: = η3: , equivalently – this can be expressed as τ1 = τ2 = τ3 – or to compare
one treatment with the average of the other treatments: for example, to compare
treatment 1 with the averages of treatments 2 and 3 (Trt1.vs.average.Trt2.and.Trt3).
For the hypothesis test of the equality of treatments (τ1 = τ2 = τ3), the estimable
function K′β is given by:
K′ =
0
0
1
0
0
1
-1
-1
0
0
0
0
0
0
; β=
η
τ1
τ2
τ3
bloq1
bloq2
bloq3
While for contrasts Trt1.vs.average.Trt2.and.Trt3 and Trt2.vs.average.Trt1.and.
Trt3, K′β is given by
Trt1:vs:average:Trt2:and:Trt3 = η1 • - 1=2ðη2 • þ η3 • Þ = τ1 -
τ2 - τ3
2
Trt2:vs:average:Trt1:and:Trt3 = η2 • - 1 =2ðη1 • þ η3 • Þ = τ2 -
τ1 - τ3
2
3.2
Illustrative Examples of the Data Scale and the Model Scale
K′ =
0 2
0 1
-1
-2
-1
1
0
0
0
0
0
; β=
0
97
η
τ1
τ2
τ3
bloq1
bloq2
bloq3
The following GLIMMIX procedure allows us to implement the above example.
proc glimmix;
class trt block;
model y = trt block/solution;
lsmeans trt / diff e;
estimate 'lsm trt1' intercept 3 trt 3 0 0 0 block 1 1 1 / divisor=3;
estimate 'overall mean' intercept 3 trt 1 1 1 1 block 1 1 1 1 / divisor=3;
estimate 'average trt1&trt2' intercept 6 trt 3 3 0 block 2 2 2 /
divisor=6;
estimate 'average trt1&trt2&trt3' intercept 9 trt 3 3 3 3 block 3 3 3 3
3/divider=9;
estimate 'trt1 vs trt2' trt 1 -1 0 ;
estimate 'trt1 vs trt3' trt 1 0 -1;
estimate 'trt2 vs trt3' trt 0 1 -1;
estimate 'trt1 vs trt2' trt 1 -1 0, 'trt1 vs trt3' trt 1 0 -1, 'trt2 vs
trt3' trt 0 1 -1/divisor=1,1,1 adjust=sidak;
contrast 'trt1 vs trt2' trt 1 -1 0 ;
contrast 'trt1 vs trt3' trt 1 0 -1;
contrast 'trt2 vs trt3' trt 0 1 -1;
contrast 'trt1 vs average trt1,trt2' trt 2 -1 -1;
contrast 'trt2 vs average trt1,trt3' trt -1 2 -1;
contrast 'type 3 trt ss' trt 1 0 -1 0,trt 0 1 -1;
contrast 'type 3 trt test' trt 2 -1 -1,trt -1 2 -1;
run;
Part of the GLIMMIX output is shown below in Table 3.9. Parameter estimation
for treatments 1–2 and blocks 1–2 are shown below, except for treatment and block
3. This is because it is an incomplete rank model. The generalized inverse is used in
the estimation through the SWEEP operator of SAS. In this case, it sets the last class
effect equal to zero (Table 3.9).
“Coefficients” (part (a) of Table 3.10) obtained with option E for the least squares
means of treatments in “lsmeans” shows how SAS uses this information in the
parameter solution to calculate the treatment means (part (b)). In part (c), we can see
the difference of means obtained with the “diff” option in “lsmeans.”
The estimates obtained from the “estimate” command with multiple estimable
functions and in the “Sidak” adjustment and contrasts are shown in Table 3.11. This
adjustment allows us to control for type I errors. The “adjust” option in “estimate” in
the Sidak adjustment (part (b)) allows us to obtain the adjusted P-values denoted as
AdjP in addition to Pr > |t|.
98
3
Objectives of Inference for Stochastic Models
Table 3.9 Estimation of treatment and block parameters
Parameter estimates
Effect
Intercept ð^ηÞ
Trt ð^τ1 Þ
Trt ð^τ2 Þ
Trt ð^τ3 Þ
^ 1Þ
BLOCK ðblock
Trt
BLOCK
1
Estimate
43.5556
7.3333
-18.0000
0
1.0000
2
-6.6667
1
2
3
^ 2Þ
BLOCK ðblock
^ 3Þ
BLOCK ðblock
3
Pr > |t|
0.0002
0.1036
0.0067
4
0.29
0.7887
4
-1.91
0.1288
DF
4
4
4
3.4907
3.4907
0
Scale σ^2
Table 3.10 Coefficients for
treatment and block used in
least squares
t-value
13.67
2.10
-5.16
Standard error
3.1866
3.4907
3.4907
18.2778
12.9243
(a) Coefficients for Trt least squares means
Effect
Trt
BLOCK
Intercept
Trt
1
Trt
2
Trt
3
Row1
Row2
Row3
1
1
1
1
1
1
BLOCK
1
0.3333
0.3333
0.3333
BLOCK
2
0.3333
0.3333
0.3333
BLOCK
3
0.3333
0.3333
0.3333
(b) Trt least squares means
Trt
Estimate
Standard error
DF
t-value
Pr > |t|
1
49.0000
2.4683
4
19.85
<0.0001
2
23.6667
2.4683
4
9.59
0.0007
3
41.6667
2.4683
4
16.88
<0.0001
(c) Differences of Trt least squares means
Trt
_Trt
Estimate
Standard error
DF
t-value
Pr > |t|
1
2
25.3333
3.4907
4
7.26
0.0019
1
3
7.3333
3.4907
4
2.10
0.1036
3
3
-18.0000
3.4907
4
-5.16
0.0067
The planned contrasts in matrix K′ and with the F-values obtained with the
“contrast command” produce the same results (part (c)).
Now, the same dataset is fitted using the same predictor but assuming that the
response variable is binomial. This analysis intends to show the options available in
the SAS commands when you want to fit non-normal responses; in this case, it is
binomial. Practically, the same commands used in the previous program with normal
data are used, but, now, some other options (“ilink,” “oddsratio,” or “exp”) are
exemplified with details under what circumstances they should be used. This is
because all estimable functions produce estimates at the model scale, and we must
3.2
Illustrative Examples of the Data Scale and the Model Scale
99
Table 3.11 Multiple estimates and contrasts
(a) Estimates
Label
Estimate
Standard error
DF
t-value
LSM Trt1
49.0000
2.4683
4
19.85
Overall mean
38.1111
1.4251
4
26.74
Average Trt1&Trt2
36.3333
1.7454
4
20.82
Average Trt1&Trt2&Trt3
38.1111
1.4251
4
26.74
Trt1 vs Trt2
25.3333
3.4907
4
7.26
Trt1 vs Trt3
7.3333
3.4907
4
2.10
Trt2 vs Trt3
-18.0000
3.4907
4
-5.16
(b) Estimate adjustment for multiplicity: Sidak
Label
Estimate
Standard error
DF
t-value
Pr > |t|
Trt1 vs Trt2
25.3333
3.4907
4
7.26
0.0019
Trt1 vs Trt3
7.3333
3.4907
4
2.10
0.1036
Trt1 vs Trt3
-18.0000
3.4907
4
-5.16
0.0067
(c) Contrasts
Label
Num DF
Den DF
F-value
Trt1 vs Trt2
1
4
52.67
Trt1 vs Trt3
1
4
4.41
Trt2 vs Trt3
1
4
26.59
Trt1 vs average Trt1,Trt2
1
4
29.19
Trt2 vs average Trt1,Trt3
1
4
51.37
Type 3 Trt SS
2
4
27.89
Type 3 Trt Test
2
4
27.89
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
0.0019
0.1036
0.0067
Adj P
0.0057
0.2796
0.0200
Pr > F
0.0019
0.1036
0.0067
0.0057
0.0020
0.0045
0.0045
decide what conversions are necessary to obtain the results at the data scale. Below,
the estimable functions and the appropriate conversion required to produce the
results on the data scale are listed.
(a) Least squares means (“lsmeans”) for normal data and an inverse link (“ilink”) for
non-normal data
(b) Difference between pairs of treatment means of “lsmeans” for normal data and
“odds ratio” for non-normal data
(c) Estimation of the mean of a treatment (“estimate”) for normal data and an inverse
link (“ilink”) for non-normal data
(d) Estimation of a treatment i vs treatment i′: exponentiation (“exp”) (or odds ratio)
(e) Multiple estimates of treatment differences as “exp” (or odds ratio) for
non-normal data
(f) In “contrast estimation,” conversion to the data scale is not necessary, since it is
only an F-statistic test.
The following GLIMMIX program shows how to implement this model with a
binomial response.
100
3
Objectives of Inference for Stochastic Models
proc glimmix;
class trt block;
model y/n = trt block/solution oddsratio;
lsmeans trt / diff e oddsratio;
estimate 'lsm trt1' intercept 3 trt 3 0 0 0 block 1 1 1 1/divider=3 ilink;
estimate 'difference trt1 vs trt2' trt 1 -1 0/exp;
estimate 'avg trt1&trt2&trt3' intercept 9 trt 3 3 3 3 block 3 3 3 3
3/divider=9;
estimate 'trt1 vs trt2' trt 1 -1 0/exp;
estimate 'trt1 vs trt3' trt 1 0 -1/exp;
estimate 'trt2 vs trt3' trt 0 1 -1/exp;
estimate 'trt1 vs trt2' trt 1 -1 0, 'trt1 vs trt3' trt 1 0 -1, 'trt1 vs
trt3' trt 0 1 -1/exp adjust=sidak;
contrast 'trt1 vs trt2' trt 1 -1 0;
contrast 'trt1 vs trt3' trt 1 0 -1;
contrast 'trt2 vs trt3' trt 0 1 -1;
contrast 'trt1 vs average trt1,trt2' trt 2 -1 -1;
contrast 'trt2 vs average trt1,trt3' trt -1 2 -1;
contrast 'type 3 trt ss' trt 1 0 -1 0,trt 0 1 -1;
contrast 'type 3 trt test' trt 2 -1 -1,trt -1 2 -1;
run;
Part of the output is shown in Table 3.12. The estimated treatment and block
parameters of the model are given in part (a) of Table 3.12; the last two effects of
both classes were restricted to zero because they are incomplete rank design matrices. In part (b), the type III tests of fixed effects and in part (c) the odds ratio
estimates are provided. Note that σ^2 does not appear in the output because the
variance of the binomial distribution is not an independent parameter.
In Table 3.12 (parts (b) and (c)), which shows the sum of the squares of fixed
effects type III as well as the odds ratio, it can be seen that only the effect of
treatments is significant but not the effect of blocks, which indicates that it is valid
to analyze these data using a completely randomized design. Two sets of odds ratios
were estimated (part (c)): one for the treatment effects and the other for the block
effects in the model. In the calculation of odds ratios, generally, the last level of the
factor is compared with the rest of the levels of that same factor.
The estimates obtained with “estimate”, in Table 3.13 (parts (a) and (b)), are
results in terms of the model scale, whereas the last column is obtained by applying
EXP ðeτi - τi 0 Þ.
The least squares means for treatment and the linear predictors of treatment
differences (parts (a) and (b) of Table 3.14, respectively) obtained with “lsmeans”
are the values under the “Estimate” column, and, these, together with their
corresponding standard errors, were obtained using the linear predictor
^ .
^ηij = ^η þ ^τi þ block
•
These estimates are on the model scale, whereas the values under the “Mean”
column and their respective standard errors were obtained by applying the inverse
link to obtain the probabilities of success of each treatment (^
π i Þ. While the estimated
linear predictors for the mean differences were obtained with the “oddsratio” option,
the mean difference in the data scale is obtained by taking the inverse of these
predictors.
3.3
Fixed and Random Effects in the Inference Space
101
Table 3.12 Results of the analysis of variance in a binomial model
(a) Parameter estimates
Effect
Trt
Intercept ð^ηÞ
Trt ð^τ1 Þ
1
Trt ð^τ2 Þ
2
3
Trt ð^τ3 Þ
^
BLOCK ðblock1 Þ
^ 2Þ
BLOCK ðblock
^ 3Þ
BLOCK ðblock
BLOCK
Pr > |t|
0.0536
0.0785
0.0059
4
0.31
0.7698
4
-0.33
0.7559
Standard error
0.1883
0.2169
0.2079
DF
4
4
4
1
0.2088
2
-0.07205
0.2164
3
0
(b) Type III tests of fixed effects
Effect
Num DF
Trt
2
BLOCK
2
(c) Odds ratio estimates
Trt
BLOCK
_Trt
_BLOCK
1
3
2
3
1
3
2
3
3.3
t-value
2.71
2.35
-5.33
Estimate
0.5099
0.5097
-1.1088
0
0.06541
Den DF
4
4
estimate
1.665
0.330
1.068
0.930
F-value
29.55
0.20
DF
4
4
4
4
Pr > F
0.0040
0.8258
95% confidence limits
0.912
3.040
0.185
0.588
0.598
1.906
0.510
1.697
Fixed and Random Effects in the Inference Space
In an analysis, inference can be directed solely at fixed effects (population inference)
or at a combination of fixed and random effects (specific inference). To illustrate
these two levels of inferences, we will consider two examples:
3.3.1
A Broad Inference Space or a Population Inference
In practice, the random effects in a linear mixed model (LMM) should represent the
population from which the data were collected and should be included in studies as if
they came from a well-planned sample. In a model, random effects can be locations,
regions, states, blocks, and so on, and they have two very particular characteristics.
• Random effects represent the target population.
• Random effects have a probability distribution.
These two characteristics allow us to have a broad inference space where we can
calculate point estimates, estimate intervals, and perform hypothesis testing applicable to the entire population represented by the random effects. Formally, an
estimate or hypothesis test based on an LMM indicates that we have a broad
(a) Estimates
Label
Estimate
Standard error
DF
t-value
LSM Trt1
1.0174
0.1603
4
6.35
Average Trt1&Trt2&Trt3
0.3080
0.08768
4
3.51
Trt1 vs Trt2
1.6184
0.2181
4
7.42
Trt1 vs Trt3
0.5097
0.2169
4
2.35
Trt2 vs Trt3
-1.1088
0.2079
4
-5.33
(b) Estimate adjustment for multiplicity: Sidak
Standard
tExponentiated
Label Estimate error
DF value Pr > |t| Adj P estimate
Trt1
1.6184 0.2181
4
7.42 0.0018 0.0053 5.0452
vs
Trt2
Trt1
0.5097 0.2169
4
2.35 0.0785 0.2175 1.6647
vs
Trt3
-1.1088 0.2079
4
-5.33 0.0059 0.0177 0.3300
Trt1
vs
Trt3
Table 3.13 Different estimates obtained with “estimate”
Pr > |t|
0.0032
0.0246
0.0018
0.0785
0.0059
Mean
0.7345
Non-est
Non-est
Non-est
Non-est
Standard error Mean
0.03127
5.0452
1.6647
0.3300
Exponentiated estimate
.
102
3
Objectives of Inference for Stochastic Models
3.3
Fixed and Random Effects in the Inference Space
103
Table 3.14 Estimated linear predictors for treatments and treatment differences with their respective inverse values
(a) Trt least squares means
Trt Estimate
Standard error DF t-value
1
1.0174 0.1603
4
6.35
-0.6011 0.1480
4
-4.06
0.5077 0.1462
4
3.47
(b) Differences of Trt least squares means
Trt
_Trt
Estimate
Standard error
1
2
1.6184
0.2181
1
3
0.5097
0.2169
-1.1088
0.2079
2
3
Pr > |t|
0.0032
0.0153
0.0255
DF
4
4
4
Mean
0.7345
0.3541
0.6243
t-value
7.42
2.35
-5.33
Standard error Mean
0.03127
0.03385
0.03429
Pr > |t|
0.0018
0.0785
0.0059
Odds ratio
5.045
1.665
0.330
inference space defined by the estimable function K′β if Z is a matrix with coefficients equal to zero; otherwise, the estimation or hypothesis test is defined by the
prediction function K′β + Z′β, which is a specific inference.
3.3.2
Mixed Models with a Normal Response
In Example 3.2, the response variable was assumed as a function of fixed effects due
to treatments and blocks, since block effects were also assumed to be fixed effects.
Now, suppose that applications of treatments were done by three different people
(blocks); then, assuming that the block effects are fixed, this would be questionable
since each person does their job according to their experience, skill, and so forth.
Clearly, there is some variability between blocks that is not due to the experiment
and this has to be removed, so the effects due to blocks must be considered random.
In this example, let us assume that the three blocks (persons) were randomly selected
from a population. Thus, the components of the model are defined as follows:
Distribution:
(a) yijj blockj ~ N(μij, σ 2)
(b) bloquej N ð0, σ2block Þ
Linear predictor: ηij = η + τi + blockj (i, j = 1, 2, 3)
Link function: ηij = μij (identity link)
Note the impact of changing the estimable function for the mean of treatments. In
Example 3.2, the estimable function was defined by Eðyi • Þ = η þ τi þ block • . Now
with the mixed model (fixed effects and random blocks), the estimable function is
104
3 Objectives of Inference for Stochastic Models
defined by E ðyi • Þ = η þ τi because E(block) = 0. Therefore, the estimable function
for the mean in each of the treatments is η + τi. In this situation, two questions arise:
• How much do the results obtained from a fixed effects model differ from those
obtained from a mixed model?
• How can we compare the two results?
The following program allows us to estimate a mixed model with a normal
response.
proc glimmix;
class trt block;
model y = trt /solution;
random block/solution;
lsmeans trt / diff e;
estimate 'lsm trt1' intercept 1 trt 1 0 0 0|block 0 0 0 0;
estimate 'lsm trt2' intercept 1 trt 0 1 0|block 0 0 0 0;
estimate 'lsm trt3' intercept 1 trt 0 0 0 1|block 0 0 0 0;
estimate 'blup trt1' intercept 3 trt 3 0 0 0|block 1 1 1 / divisor=3;
estimate 'blup trt2' intercept 3 trt 0 3 0|block 1 1 1 / divisor=3;
estimate 'blup trt3' intercept 3 trt 0 0 3|block 1 1 1 / divisor=3;
run;
In the previous SAS GLIMMIX code, the “estimate” command shows the
coefficients associated with the fixed effects before the vertical bar (j) and after the
vertical bar, are provided the coefficients for the random effects associated with the
model, that is:
efectosfijos
1
K ′ β þ Z′ b = 1
1
1
0
0
0
1
0
efectosaleatorios
0
0
1
η
τ1
τ2
τ3
1
þ 1
1
1
1
1
1
1
1
block1
block2
block3
Part of the output is shown in Table 3.15. Subsection (a) shows the estimated
variance components due to blocks, and for conditional observations, the effect of
^2block = 11:2778, whereas the mean squared error (MSE) is
the blocks is σ
2
σ^ = 18:2778. On the other hand, the fixed effects solution obtained with the
“solution” option of the parameters is provided in part by (b). The analysis of
variance (part (c)) indicates that there is a significant difference between treatments
(P = 0.0045), and so the null hypothesis must be rejected (H0 : μ1 = μ2 = μ3).
The means estimated with the “estimate” statement, given that the mean block
effect is zero, the mean and the best linear unbiased predictor for each treatment are
similar, as shown in Table 3.16 (part (a)). Subsections (b) and (c) show the means
and differences between two means estimated with the “lsmeans” statement and the
“diff” option.
3.4
Marginal and Conditional Models
105
Table 3.15 Variance components, fixed effects, and fixed effects test
(a) Covariance parameter estimates
Cov Parm
BLOCK
Residual
(b) Solutions for fixed effects
Effect
Trt
Estimate
Intercept
41.6667
Trt
1
7.3333
Trt
2
-18.0000
Trt
3
0.
(c) Type III tests of fixed effects
Effect
Num DF
Trt
2
3.4
Estimate
11.2778
18.2778
Standard error
3.1388
3.4907
3.4907
..
Den DF
4
Standard error
17.8966
12.9243
DF
2
4
4
.
t-value
13.27
2.10
-5.16
F-value
27.89
Pr > |t|
0.0056
0.1036
0.0067
Pr > F
0.0045
Marginal and Conditional Models
The process of analyzing a dataset has two main objectives: the first is model
selection, which aims to find well-fitting parsimonious models for the responses
being measured, and the second is model prediction, where estimates from the
selected models are used to predict quantities of interest and their uncertainties.
The differences that may arise in this analysis process are mainly due to the
choice of unidentifiable constraints on random effects. To compare two different
models, we must compare analogous quantities. Different constraints can lead to
apparently extremely different but inferentially identical models. The conditional
model is believed to be the basic model, and any conditional model leads to a
specific marginal model. Lee and Nelder (2004) proposed and worked on conditional models derived from generalized hierarchical linear models (GHLMs) and
marginal models derived from these conditional models. Marginal models have
often been fitted using generalized estimating equations (GEEs), the drawbacks of
which are also discussed.
3.4.1
Marginal Versus Conditional Models
Consider two models with a normal distribution: one is a random effects model
(a mixed model)
yij = μ þ τi þ bj þ εij
where bj N 0, σ 2b and εij~N(0, σ 2). The other is a marginal model
106
3
Objectives of Inference for Stochastic Models
Table 3.16 Estimated means, best linear unbiased estimates (BLUEs), and BLUPs for treatment
and the difference between two means
(a) Estimates
Label
Estimate
Standard error
lsm trt1
49.0000
3.1388
lsm trt2
23.6667
3.1388
lsm trt3
41.6667
3.1388
blup trt1
49.0000
2.4683
blup trt2
23.6667
2.4683
blup trt3
41.6667
2.4683
(b) Trt least squares means
Trt
Estimate
Standard error
1
49.0000
3.1388
1
23.6667
3.1388
2
41.6667
3.1388
(c) Differences of Trt least squares means
Trt
_Trt
Estimate
Standard error
1
2
25.3333
3.4907
1
3
7.3333
3.4907
-18.0000
3.4907
2
3
DF
4
4
4
4
4
4
t-value
15.61
7.54
13.27
19.85
9.59
16.88
DF
4
4
4
t-value
15.61
7.54
13.27
DF
4
4
4
t-value
7.26
2.10
-5.16
Pr > |t|
<0.0001
0.0017
0.0002
<0.0001
0.0007
<0.0001
Pr > |t|
<0.0001
0.0017
0.0002
Pr > |t|
0.0019
0.1036
0.0067
E yij = μ þ τi
where the elements in V( y) = Σ are variances and covariances that have an arbitrary
correlation structure. Zeger et al. (1988) pointed out that given a marginal model, the
generalized estimating equations are consistent. An obvious advantage of using
random effects models is that they allow conditional inferences in addition to
marginal inferences (Robinson 1991). Using the model with random effects, we
can obtain not only the conditional mean
μCij = E yij jbj = μ þ τi þ bj
but also the marginal mean
μij = E μCij = E yij jbj = μ þ τi
whereas with the marginal model, we can obtain only the marginal mean μij.
It may be reasonable to assume that the unobservable characteristic of the random
effects of blocks (bj) follows a certain distribution. However, the center of this
distribution cannot be identified because it is confounded with the intercept. Therefore, in the random effects model, we put the unidentifiable constraints E(bi) = 0 and
E(εij) = 0 as we do for error terms in linear models. In the mixed model, these
^
restrictions are
εij = 0 in any estimation procedure. First, we
j bj = 0 and
j^
3.4
Marginal and Conditional Models
107
Table 3.17 Mortality of coffee seedling clones (C) in different substrates (S)
Block
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
S
3
3
3
1
1
1
2
2
2
4
4
4
4
4
4
1
1
1
2
2
2
3
C
1
3
2
1
3
2
1
3
2
1
3
2
3
1
2
3
1
2
3
1
2
3
Mortality
3.33
16
16
3.33
6.6
3.3
10
3.33
3.33
3.33
16
13.3
3.3
3.3
20
10
3.33
6.6
36.6
26.6
43.3
3.3
Pct
0.0333
0.16
0.16
0.0333
0.066
0.033
0.1
0.0333
0.0333
0.0333
0.16
0.133
0.033
0.033
0.2
0.1
0.0333
0.066
0.366
0.266
0.433
0.033
Block
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
S
3
3
3
2
2
2
4
4
4
1
1
1
4
4
4
2
1
1
2
2
2
C
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
3
2
3
1
2
3
Mortality
6.6
10
56.6
3.3
26.6
40
3.3
46
33.3
6.6
43.3
50
33
10
23.3
50
23.3
6.6
16
10
16
Pct
0.066
0.1
0.566
0.033
0.266
0.4
0.033
0.46
0.333
0.066
0.433
0.5
0.33
0.1
0.233
0.5
0.233
0.066
0.16
0.1
0.16
consider the case in which the data follow a normal distribution. We then briefly
discuss how the results differ for data with a non-normal distribution.
3.4.2
Normal Distribution
Example The effect of different substrates (factor A), i.e., three substrates made
from vermicompost and one from compost, on the development of physiological
variables and mortality of cuttings of three clones (factor B) of robusta coffee
(Coffea canephora p.) was evaluated. The levels of factor A are randomly assigned
to rows in each block, with the following restriction: each block receives levels A1,
A2, A3, and A4 and each level of factor B (B1, B2, and B3) is randomly assigned to
each level of factor A in each block. The data for this experiment are tabulated in
Table 3.17.
Note that while there are two randomization processes, there are effectively three
sizes of experimental units: rows for A levels, columns for B levels, and row–column
intersections for A × B combinations. Thus, the experimental design used was a
complete randomized design with a strip-plot treatment arrangement.
108
3
Objectives of Inference for Stochastic Models
The model, for these data, is given below:
yijk = μ þ bk þ αi þ ðαbÞik þ βj þ ðβbÞjk þ ðαβÞij þ εijk
where yijk is the kth response observed at the ith level of factor A and at the jth level
of factor B, μ is the overall mean, bk is the random effect due to blocks assuming
bk N 0, σ 2b , αi is the fixed effect due to substrate type (S), (αb)ik is the random
effect due to the interaction of a substrate with blocks assuming ðαbÞik N 0, σ 2αb ,
βj is the fixed effect due to the coffee clone type (C), (βb)jk is the random effect due to
the interaction of a coffee clone with blocks assuming ðβbÞjk N 0, σ 2βb , (αβ)ij is
the interaction fixed effect between a substrate and a coffee clone, and εijk is the
normal random error εijk~N(0, σ 2). The components of the model for this dataset are
as follows:
Linear predictor: ηijk = μ + bk + αi + (αb)ik + βj + (βb)jk + (αβ)ij
Distributions: yijk j bk, (αb)ik, (βb)jk~N(μijk, σ 2)
bk N 0, σ 2b ; ðαbÞik N 0, σ 2αb ; ðβbÞjk N 0, σ 2βb
Link function: ηijk = μijk
The following GLIMMIX syntax sets a GLMM with a normal response.
proc glimmix;
class block s c;
model y=s|c;
random intercept s w/subject=block;
lsmeans s*c/ slicediff=s;
run;
Part of the results of this analysis is shown below. The estimated variance
components for blocks, block × substrate, blocks × clon, and the MSE are
σ^2b = 23:4714, σ^2αb = 35:4995, σ^2βb = 67:0160 and σ^2 = CME = 139:58, respectively,
which are listed in part (a) of Table 3.18. However, the fixed effects tests for both
factors and the interaction (part (b)) are not statistically significant.
According to the “slicediff = s” option in the “lsmeans” statement, Table 3.19
shows the simple effects of each substrate level at varying clone levels.
3.4.3
Non-normal Distribution
Example Using the data in Table 3.17 but under a beta distribution, the components
of the GLMM change slightly:
3.4
Marginal and Conditional Models
Table 3.18 Estimated variance components and type III
tests of fixed effects
109
(a) Covariance parameter estimates
Cov Parm
Subject
Estimate
Standard error
Intercept
Block
23.4714
58.9336
S
Block
35.4995
44.9134
C
Block
67.0160
64.0909
Residual
139.58
49.9056
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
S
3
8.318
0.48
0.7076
C
2
5.935
1.68
0.2650
S*C
6
16.65
0.44
0.8441
Table 3.19 Simple effect comparisons across substrate levels
Simple effect comparisons of S*C least squares means by S
Simple effect level
C
_C
Estimate
Standard error
S1
1
2
-12.9814
10.9667
S1
1
3
-12.1564
10.9667
S1
2
3
0.8250
10.1636
S2
1
2
-6.8325
10.1636
S2
1
3
-15.2779
9.8708
S2
2
3
-8.4454
9.8708
S3
1
2
-4.4620
12.8219
S3
1
3
-19.0350
12.8124
S3
2
3
-14.5730
11.4417
S4
1
2
-11.5925
10.1636
1
3
-8.2425
10.1636
S4
S4
2
3
3.3500
10.1636
DF
15
15
15
15
15
15
15
15
15
15
15
15
t- value
-1.18
-1.11
0.08
-0.67
-1.55
-0.86
-0.35
-1.49
-1.27
-1.14
-0.81
0.33
Pr > |t|
0.2549
0.2851
0.9364
0.5116
0.1425
0.4057
0.7327
0.1581
0.2222
0.2719
0.4301
0.7463
Distributions: yijk j bk, (αb)ik, (βb)jk~Beta(μijk, ϕ), where ϕ is the scale parameter.
bk N 0, σ 2b ; ðαbÞik N 0, σ 2αb ; ðβbÞjk N 0, σ 2βb
Linear predictor: ηijk = μ + bk + αi + (αb)ik + βj + (βb)jk + (αβ)ij
Link function: ηijk = logit(μijk)
The following GLIMMIX syntax sets a beta response variable.
proc glimmix method=laplace;
class block s c;
model pct=s|c/dist=beta;
random intercept s w/subject=block;
lsmeans s*c/plot=meanplot(sliceby=s join) slicediff=s ilink;
run;
110
Table 3.20 Variance components and the fixed effects test
3
Objectives of Inference for Stochastic Models
(a) Covariance parameter estimates
Cov Parm
Subject
Estimate
Intercept
block
0.06723
S
block
0.1594
C
block
0.1932
^
16.6041
Scale ϕ
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
S
3
8
C
2
6
S*C
6
15
Standard error
0.1617
0.1420
0.1687
5.6153
F-value
0.74
2.60
0.59
Pr > F
0.5584
0.1540
0.7303
Table 3.21 Simple effect comparisons across substrate levels
Simple effect comparisons of S*C least squares means by S
Simple effect level
C
_C
Estimate
Standard error
S1
-0.9698
0.6578
1
2
S1
-0.9298
0.6696
1
3
S1
2
3
0.03999
0.5457
S2
-0.4407
0.5432
1
2
S2
-0.8555
0.5237
1
3
S2
-0.4149
0.4911
2
3
S3
-0.4588
0.8285
1
2
S3
-1.3224
0.7773
1
3
S3
-0.8636
0.6375
2
3
S4
-0.9880
0.5619
1
2
S4
-0.7138
0.5739
1
3
S4
2
3
0.2741
0.5261
DF
15
15
15
15
15
15
15
15
15
15
15
15
t-value
-1.47
-1.39
0.07
-0.81
-1.63
-0.84
-0.55
-1.70
-1.35
-1.76
-1.24
0.52
Pr > |t|
0.1611
0.1852
0.9426
0.4299
0.1231
0.4115
0.5879
0.1095
0.1955
0.0991
0.2326
0.6099
Some of the SAS output from this analysis is shown below. The variance
components estimated for blocks, block × substrate, blocks × clon, and the scale
parameter are σ^2b = 0:06723, σ^2αb = 0:1594, σ^2βb = 0:1932, and with scale parameter
^ = 16:6041, respectively, which are listed in part (a) of Table 3.20. However, the
ϕ
fixed effects tests for both factors and the interaction (part (b)) are not statistically
significant. Unlike a normal distribution (the previous example), the variance components (multiplied by 100) under the beta distribution are smaller, and the type III
fixed effects test is closer to be significant.
Table 3.21 shows, for each substrate level at varying clone levels, the estimates
(linear predictors) of the simple effects. These effects differ from the previous
results, but this is mainly because in a GLMM, these values correspond to the linear
predictors estimated at the model scale and not to the estimated means at the data
3.5
Exercises
111
Table 3.22 Total yields (grams) of barley varieties in 12 independent trials
Location
1
1
2
2
3
3
4
4
5
5
6
6
Variety
Manchuria
81.0
80.7
146.6
100.4
82.3
103.1
119.8
98.9
98.9
66.4
86.9
67.7
Svansota
105.4
85.3
142.0
115.5
77.3
105.1
121.4
61.9
89.0
49.9
77.1
66.7
Velvet
119.7
80.4
150.7
112.2
78.4
116.5
124.0
96.2
69.1
96.7
78.9
67.4
Trebi
109.7
87.2
191.5
147.4
131.3
139.9
140.8
125.5
89.3
61.9
101.8
91.8
Peatland
98.3
84.2
145.7
108.1
896
129.6
124.8
75.7
104.1
80.3
96.0
94.1
scale (Example 3.4.2). It is also important to note that the degrees-of-freedom
correction in the estimation of means cannot yet be used in the estimation of a
GLMM.
3.5
Exercises
Exercise 3.5.1 The data in the Table 3.22 below show the yield of five barley
varieties in a randomized complete block experiment conducted in Minnesota
(Immer et al. 1934).
• Write a complete description of the statistical model associated with this study
and the assumptions of this model.
• Compute the ANOVA for the design model according to part (a) and determine
whether there is a significant difference in the varieties.
• Use the least significance difference (LSD) method to make pairwise comparisons of variety mean yields.
Exercise 3.5.2 Lew (2007) conducted an experiment to determine whether cultured
cells respond to two drugs (chemical formulations). The experiment was conducted
using a cell culture line placed in Petri dishes. Each experimental trial consisted of
three Petri dishes: one treated with drug 1, one treated with drug 2, and one untreated
as a control. The data are shown in the following Table 3.23:
(a) Write a complete description of the statistical model associated with this study
and the assumptions of this model.
(b) Analyze the data using a completely randomized design. Is there a significant
difference between the treatment groups?
112
Table 3.23 Number of cells
cultured in different drugs
3
Ensayo 1
Ensayo 2
Ensayo 3
Ensayo 4
Ensayo 5
Ensayo 6
Objectives of Inference for Stochastic Models
Control
1147
1273
1216
1046
1108
1265
Drug 1
1169
1323
1276
1240
1432
1562
Drug 2
1009
1260
1143
1099
1385
1164
(c) Analyze the data as a randomized complete block design, where the number of
trials represents a blocking factor.
(d) Is there any difference in the results obtained in (a) and (b)? If so, explain what
might be the cause of the difference in results and what method would you
recommend?
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Chapter 4
Generalized Linear Mixed Models
for Non-normal Responses
4.1
Introduction
Generalized linear mixed models (GLMMs) have been recognized as one of the
major methodological developments in recent years, which is evidenced by the
increased use of such sophisticated statistical tools with broader applicability and
flexibility. This family of models can be applied to a wide range of different data
types (continuous, categorical (nominal or ordinal), percentages, and counts), and
each is appropriate for a specific type of data. This modern methodology allows data
to be described through a distribution of the exponential family that best fits the
response variable. These complex models were not computationally possible up until
recently when advances in statistical software have allowed users to apply GLMMs
(Zuur et al. 2009; Stroup 2012; Zuur et al. 2013). Researchers in fields other than
statistical science are also interested in modeling the structure of data. For example,
in the social sciences there have been applications in the field of education when
several tests are applied to students; in longitudinal personality studies when the
occurrence of an emotion is repeatedly observed over time over a set of people; and
in surveys to investigate the political preference of a population, among others.
Likewise, agriculture and life sciences are other major areas, where the measurement of response variables depart from the conventionally used classical methodology based on “normality” to model or describe the data set, i.e., data that generally
fall within the nominal, ordinal or interval (continuous) scales of measurement. In a
GLMM, the data response does not undergo any transformation, but, instead, the
response is modeled as a function of the expected value through a linear relationship
with the explanatory variables. GLMMs, a powerful tool, allow proper modeling of
variations between groups and between space and time, leading to accuracy in the
modeling of the observed data as well as in the estimation of variance components.
© The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8_4
113
114
4.2
4
Generalized Linear Mixed Models for Non-normal Responses
A Brief Description of Linear Mixed Models (LMMs)
Before addressing GLMMs, we present a brief overview of linear mixed models
(LMMs). An LMM is a model whose response variable is normal and assumes:
(1) that the relationship between the mean of the dependent variable ( y) and fixed
and random effects can be modeled as a linear function; (2) that the variance is not a
function of the mean; and (3) that random effects follow a normal distribution.
The classic representation of an LMM in the matrix form, is
y = Xβ þ Zb þ ε
ð4:1Þ
where y is the vector (n × 1) of the response variable; X is the design matrix
(n × ( p + 1)) of fixed effects with rank k; β is the vector of unknown parameters
(( p + 1) × 1); Z is the design matrix (n × q) of random effects; and b is the vector of
unknown parameters of random effects (q × 1), assuming that the vector of random
effects b follows a normal distribution with mean 0 and variance matrix G, that is,
b~N(0, G). Finally, ε is the error vector with a normal distribution with mean 0 and a
variance–covariance matrix (ε~N(0, R)); both vectors b and ε are assumed to be
independent of each other.
Model 4.1, as previously mentioned, can be described in terms of a probability
distribution in two ways: the first is the marginal model y~N(E[y] = Xβ, Var
[y] = V = ZGZ′ + R), where the mean is based solely on the fixed effects, and the
parameters describing the random effects are contained in the variance and covariance matrix V (Littell et al. 2006), while the second form is the conditional model
y j b~N(Xβ + Zb, R). Under normality assumptions, both models are exactly the
same and hence produce the same solution, whereas when normality is not satisfied,
the models produce different solutions (Stroup 2012).
4.3
Generalized Linear Mixed Models
Most datasets in agricultural, biological, and social sciences often fall outside the
scope of the traditional methods taught in introductory statistics and statistical
methods. Often, these data (response variables) are: (a) binary (the presence or
absence of a trait of interest, success or failure, the infection status of an individual,
or the expression of a genetic disorder); (b) proportional (the ratio of females to
males, infection or mortality rates within a group of individuals); or (c) counts (the
number of emerging seedlings, the number of sprouts, etc.), where basic statistical
methods attempt to quantify the effects of each predictor variable. However, often,
studies of these experiments involve random effects, the purpose of which is to
quantify variation among individuals or units. The most common random effects are
blocks in experimental or observational studies that are replicated across sites
(locations or environments) or over time. Random effects also encompass variations
4.3
Generalized Linear Mixed Models
115
among individuals (when measuring multiple responses per individual such as
survival of multiple offspring or sex ratios of multiple offspring), genotypes, species,
and regions or periods over time.
GLMMs are a powerful class of statistical tools that combine the concepts and
ideas of generalized linear models (GLMs) with linear mixed models (LMMs). That
is, a GLMM is an extension of the GLM, in which the linear predictor contains
random effects in addition to fixed effects. These models handle a wide range of both
response distributions and scenarios in which observations are sampled. GLMMs
extend the theory of LMMs to response variables that have a non-normal distribution. In GLMMs, the response data are not transformed; instead, the explanatory
variables are expressed as a linear relationship through a function g of the expectation of y j b; that is, the response is conditional on random effects. This performs the
link function that relates the response to the explanatory variables in a linear manner,
thus allowing the use of standard LMM techniques for estimation and hypothesis
testing.
A conditional model is used to describe a GLMM with non-Gaussian errors
(Model 4.1), given a link function (g), as shown below:
gðηÞ = Xβ þ Zb,
which is a function of the conditional expectation given by
E ½yjb = g - 1 ðXβ þ ZbÞ = g - 1 ðηÞ = μ
ð4:2Þ
where g-1(∙) is the inverse link and the other terms have already been mentioned
earlier. The fixed and random effects are combined to form the conditional linear
predictor
η = gðE ½yjbÞ = Xβ þ Zb
ð4:3Þ
The relationship between the linear predictor and the vector of observations is
modeled as follows:
y j b g - 1 ðηÞ, R
ð4:4Þ
The above notation (4.4) expresses the conditional distribution of y, given b has a
mean g-1(η) and variance R. Note that instead of specifying the distribution for y as
in the case of a GLM, we specify a distribution for the conditional response y j b.
The variance and covariance matrix for the observations is given by:
V ðyÞ = E ½V ðyjbÞ þ V ðE½yjbÞ = A1=2 RA1=2 þ ZGZ0
ð4:5Þ
where matrix A is a diagonal matrix containing the variance functions of the model.
GLMMs cover an important group of statistical models, such as:
116
4
Generalized Linear Mixed Models for Non-normal Responses
(a) Linear models (LMs): absence of random effects, identity link function and the
assumption of a normal distribution.
(b) Generalized linear models (GLMs): random effects are absent, link function is
different from the identity function, and the response variables are non-normally
distributed.
(c) Linear mixed models (LMMs): presence of random effects, identity link function
and normal distribution assumed for the response variable.
GLMMs have been formulated to correct the shortcomings of LMMs, as there are
many cases where the assumptions made in linear mixed models are inadequate.
First, an LMM assumes that the relationship between the mean of the dependent
variable (y) and fixed and random effects (β, b) can be modeled through a linear
function. This assumption is questionable, like when a researcher wishes to model
the incidence of a disease or the success or failure of an event.
The second assumption of an LMM is that variance is not a function of the mean
and that the random effects follow a normal distribution. The assumption of constant
variance is not met when the response variable is binary (1, 0). In this case, the
variance is π(1 - π), which is a function of the mean. The result is a random variable,
which can take two values (0, 1); in contrast, the normal distribution can take any
real number. Finally, the predictions for an LMM can take any real value, whereas
the predictions for a binary variable are bounded in the interval (0, 1), since it is a
probability and this prediction cannot support negative values.
Historically, a number of options have been used to address and solve some
LMM problems, even though their use is not the most appropriate. These include
applying logarithmic transformations (log( y)), transformations using the square root
p
y , arcsine transformations (seno-1( y)), and so on. However, many of these
transformations use linear mixed models by ignoring the fact that these models are
not the most accurate, despite being aware that the response variable does not satisfy
the assumption of normality. These options are attractive because they are relatively
simple and easy to implement using the LMM machinery. However, they circumvent the problem that a linear mixed model is not the best model for analyzing data.
4.4
The Inverse Link Function
In a GLMM, the canonical link function maps the original data to the linear predictor
of the model g(η) = Xβ + Zb. This linear predictor can be transformed to an observed
data scale through an inverse link function. In other words, the inverse link function
is used to map the value of the linear predictor for the ith observation to the
conditional mean at the data scale ηi. For example, suppose that we are conducting
an experiment in which we are assessing the number of undesirable weeds observed
in a crop of interest after the application of a certain number of treatments; the
response variable is assumed to have a Poisson distribution with a mean λij, the linear
predictor of which is given by
4.6
Specification of a GLMM
117
ηij = η þ τi þ bj
where η is the intercept, τi is the fixed effect due to treatments, and bj is the random
effect assuming bj N 0, σ 2b .
To obtain the inverse function of the following predictor
log λij = g ηij = η þ τi þ bj ,
we proceed by exponentiating both sides of the previous equation, with which we
obtain the inverse function of the link shown below:
λij = eηþτi þbj ,
which is denoted as g-1(g(ηij)) = g-1(η + τi + bj).
Therefore, for this example, λij depends on the linear predictor through the inverse
link function and the variance σ 2ij depends on λij through the variance function.
4.5
The Variance Function
The variance function is used to model the inconsistent variability of the phenomenon under study. With GLMMs, the residual variability arises from two sources,
namely, the variability of the distribution of sampling units in an experimental
arrangement (blocks, plots, locations, etc.) and the variability due to overdispersion.
Overdispersion can be modeled in several ways. When dealing with a GLMM, the
scale parameter or the dispersion parameter ϕ is extremely important since it can
either increase or decrease the variance in the model for each observation.
Var yij jbj = ϕVar ηij
If overdispersion exists, one way to remove it is to add the random effects (in SAS
_residual_) of each observation to the linear predictor. Another alternative is to use
another distribution to model the dataset; for example, the two-parameter negative
binomial (NB) distribution (ηij, ϕ) instead of the single-parameter Poisson distribution (λij) in the case of count data.
4.6
Specification of a GLMM
A GLMM is composed of three parts: (1) fixed effects that convey systematic and
structural differences in responses; (2) random effects that convey stochastic differences between blocks or other random factors, as these effects allow generalizations
118
4
Generalized Linear Mixed Models for Non-normal Responses
Table 4.1 Common distributions with their respective link functions
Distribution
Binomial
Poisson
Beta
Normal
Negative
binomial
Multinomial
Link function
Logit or probit
Log
Logit
Identity
Log
Cumulative
logit
Distribution syntax
dist = binomial|bin|b
dist = Poisson|Poi
dist = beta
dist = normal|gausian
dist = negbinomial|negbin|
nb
dist = multinomial|multi
Syntax of the link
function
link = logit or probit
link = log
link = logit
link = identity|id
link = log
link = cumlogit|clogit
of the population from which the sampling units have been (randomly) sampled; and
(3) distribution of errors. Thus, a complete definition of a GLMM is as follows:
y j b f ðμ, ϕÞðconditional distributionÞ
b N ð0, GÞðrandom effectsÞ
gðμÞ = ηðlink functionÞ
η = Xβ þ Zbðlinear predictorÞ
where the distribution function f(∙) is a member of the exponential family, g(μ) is the
linear function, X and Z are the design matrices, and β and b are the unknown
parameters for fixed and random effects, respectively.
When fitting a GLMM, the data remain on the original measurement scale (data
scale). However, when means are estimated from a linear function of the explanatory
variables (the predictor), these means are on the model scale. A link function is used
to link the model scale back to the original data scale. This is not the same as
transforming the original measurements to a different measurement scale. For
example, applying the log transformation for counts followed by an analysis of
variance (ANOVA) under a normal distribution is not the same as fitting a generalized linear model, assuming a Poisson distribution and using a log link (Gbur et al.
2012). In the first case, the least squares means would normally be equal to the
arithmetic means, whereas in the second case, the means are inversely linked to the
data scale, which may not be equal to the arithmetic means of the original sample.
The distribution specifications in “proc GLIMMIX” have default link functions,
but it is always highly recommended to explicitly code the link function, since for
some type of response variable, more than one alternative exists. This way, there is
no doubt that an appropriate function was used. Using the wrong link function will
lead to totally meaningless and incorrect results. Table 4.1 shows some common
distributions, the appropriate link function, and the proper syntax for each.
For a complete list, see the online Statistical Analysis Software (SAS/STAT)
documentation for PROC GLIMMIX.
4.7
Estimation of the Dispersion Parameter
4.7
119
Estimation of the Dispersion Parameter
The overall measures of fit compare the observed values of the response variable
with fitted (predicted) values. The dispersion parameter is unknown and therefore
must be estimated. There are two methods for estimating the overdispersion parameter. McCullagh (1983) proposed estimating overdispersion as follows:
ϕ =
ðy - μÞ0 V μ- 1 ðy - μÞ Pearson0 s χ 2
=
N -p
N -p
where V μ- 1 is the diagonal matrix of the variance functions and N - p is the degree
of freedom for lack of fit. Later, McCullagh and Nelder (1989) suggested using
deviance
ϕ=
- 2½lnðLM 1 Þ - lnðLðM 2 ÞÞ
Deviance
=
N -p
N -p
Deviance is a global fit statistic that also compares fitted and observed values;
however, its exact function depends on the likelihood function of the random
component of the model. Deviance compares the maximum value of the likelihood
function of a model, like M1, with the maximum possible value of the likelihood
function that is calculated using data. When data are used in the likelihood function,
the model is saturated and has as many parameters as possible. Thus, M2 is saturated
and has as many parameters as the data. Model M2 tries to fit the data and gives the
highest possible value for the likelihood.
If the overdispersion parameter is significantly greater than one, this indicates that
overdispersion exists; in other words, it indicates that the variance is greater than the
mean. Therefore, the parameter should be used to adjust the variance. If
overdispersion is not taken into account, inflated test statistics may be generated.
However, when the dispersion parameter is less than 1, the test statistics are more
conservative, which is not considered a big problem.
The following example is intended to show how GLIMMIX in SAS estimates the
dispersion parameter in a GLMM.
Example An agronomist wants to test the effectiveness of a new herbicide offered
on the market (we will denote this as herb_N) and compare it with the herbicide that
has been used for several cycles (herb_C). The experimental arrangement used was a
randomized complete block design as shown below (Table 4.2).
The components of a GLMM with a Poisson response variable are listed below:
Distribution: yij j bj Poisson λij
bj N 0, σ 2bloque
120
4
Table 4.2 Number of undesirable weeds per plot
Generalized Linear Mixed Models for Non-normal Responses
Block
1
2
3
4
5
6
7
8
Herb_C
1
5
21
7
2
6
0
19
Herb_N
36
109
30
48
3
0
5
26
Linear predictor: ηij = η þ herbicidei þ bj
Link function: log λij = ηij
This model assumes that the slopes are the same for each herbicide. The following
SAS code is used for the proposed model:
proc glimmix nobound method=laplace;
class block trt;
model count = trt/dist=poisson link=log;
random block;
lsmeans trt/ilink lines;
run;
Explanation The “method = ”option is used to specify the method used to optimize the logarithm of the likelihood function. In “proc GLIMMIX,” there are two
popular methods: adaptive quadrature (quad) or Laplace (laplace), which are the
preferred methods for categorical response variables. Both of these methods fit a
conditional model. When the quadrature method is used (method = quad), subjects
(individuals) must be declared in the random effects (e.g., for the above program,
“random intercept/subject=block”). In addition, processing random effects by subject is more efficient than using the syntax “random block” random effects in blocks.
The “dist” option is where you specify the probability distribution that is appropriate
for the type of response; in this case, it is the Poisson distribution. The “link” option
is for specifying the link function of the distribution. The “ddfm” option is omitted
so that GLIMMIX uses – by default – the method for calculating the denominator
degrees of freedom for the fixed effects tests that result from the model. The “ilink”
option converts the estimates of the treatment means (lsmeans) on the model scale to
the data scale. Finally, “proc GLIMMIX” supports the “lines” option, which adds
letter groups to the mean differences resulting from using “lsmeans.”
The most relevant parts of the SAS output, for the purposes of what we want to
show, are shown in Tables 4.3 and 4.4. The fit statistics of the fitted model are shown
in part (a) and part (b) of Table 4.3. The -2 log likelihood statistic is extremely
4.7
Estimation of the Dispersion Parameter
Table 4.3 Fit statistics and
variance components
121
(a) Fit statistics (Akaike’s information criterion (AIC), a small
sample bias corrected Akaike’s information criterion
(AICC), Bozdogan Akaike’s information criterion (CAIC),
Schwarz’s Bayesian information criterion (BIC), Hannan and
Quinn information criterion (HQIC))
-2 Log likelihood
175.35
AIC (smaller is better)
181.35
AICC (smaller is better)
183.35
BIC (smaller is better)
181.59
CAIC (smaller is better)
184.59
HQIC (smaller is better)
179.74
(b) Fit statistics for conditional distribution
139.03
-2 Log L (count | r. effects)
Pearson’s chi-square
77.56
Pearson’s chi-square/degree of freedom (DF)
4.85
(c) Covariance parameter estimates
Cov Parm
Estimate
Standard error
Block
1.5590
0.8690
Table 4.4 Type III fixed effects tests and estimated least squares means
(a) Type III tests of fixed effects
Effect
Num DF
Herbicide
1
(b) Trts least squares means
Trts
Estimate Standard error
Herb_C 1.4604
0.4696
Herb_N 2.8947
0.4561
Den DF
7
DF
7
7
t-value
3.11
6.35
F-value
101.34
Pr > |t|
0.0171
0.0004
Mean
4.3076
18.0778
Pr > F
<0.0001
Standard error mean
2.0227
8.2447
useful for comparing nested models, whereas the different versions of information
criteria that exist, such as Akaike information criterion (AIC), Akaike’s information
criteria with small sample bias correction (AICC), Bayesian information criterion
(BIC), Bozdogan Akaike’s information criteria (CAIC), and Hannan and Quinn
information criteria (HQIC), are useful when comparing models that are not necessarily nested (subsection (a)). The table of fit statistics for the conditional distribution
shows the sum of the independent contributions to the conditional (part (b)) -2 log
likelihood, the value of which is 139.03, whereas the value of Pearson’s statistic
divided by the degrees of freedom for the conditional distribution (Pearson′s chi square/DF) is 4.85.
The estimated dispersion parameter (ϕ = Pearson’s chi-square/DF) has a value
far from 1; in this case, it is ϕ = 4:85, which indicates that there is a strong
overdispersion. This may be because the specified distribution of the data is not
appropriate, the counts are too small, or the variance function was not correctly
122
4 Generalized Linear Mixed Models for Non-normal Responses
specified. The estimate of the variance component due to a block is tabulated in part
(c) of Table 4.2, the estimated value of which is σ 2bloque = 1:559.
The fixed effects test and least squares means are shown in Table 4.4. The type III
fixed effects tests indicate that there is a highly significant difference (part (a)) in the
effectiveness of herbicides in weed suppression; the estimated means with their respective standard errors are tabulated under the “Mean” column (part (b)). The “Estimate”
column containing the estimates of the means of lsmeans is on the model scale. They
are derived from the log likelihood function. SAS always lists the means obtained with
lsmeans from the model scale when creating least squares means test tables. The
“Mean” column has been converted back to the data scale using the “ilink” inverse
link function. These values are estimates of the average counts for each treatment level
(in this case, the herbicide type on the data scale). When we report the results, we must
replace the corresponding model’s least squares values in the test tables with these
estimates (means on the data scale corresponding to the values in the “Mean” column).
Since there is a strong overdispersion ϕ > 1 , assuming that the data have a
Poisson distribution is risky because this implies that the mean and variance are
equal, which is an assumption implying that the data have a Poisson distribution, i.e.,
that the mean and variance are the same. A useful alternative distribution might be a
negative binomial distribution; this distribution has a mean λ and variance λ + λϕ2
with ϕ > 0 commonly known as the scale parameter.
The following is the specification of the components of a GLMM with a negative
binomial (NB) response variable:
Distribution : yij j bj Negative binomialðλij , ϕÞ
bj Nð0, σ2bloque Þ
Linear predictor: ηij = η þ herbicidei þ bj
Link function: log λij = ηij
The GLIMMIX procedure also allows modeling a GLMM with a negative
binomial response variable:
proc glimmix data=itam nobound method=laplace;
class block trts;
model count = trts/dist=negbin;
random block;
lsmeans trts/ilink;
run;
Part of the output is shown in Table 4.5. The fit statistics for the model comparison (part (a)) and that for the conditional distribution (part (b)) are both provided by
the GLIMMIX procedure when a conditional distribution is specified. Since in the
previous analysis, it was observed that overdispersion exists when assuming a
Poisson distribution, the results – under a negative binomial distribution – indicate
that this overdispersion problem no longer exists; i.e., the binomial distribution is no
4.8 Estimation and Inference in Generalized Linear Mixed Models
Table 4.5 Fit statistics
under a negative binomial
distribution
123
(a) Fit statistics
-2 Log likelihood
120.50
AIC (smaller is better)
128.50
AICC (smaller is better)
132.13
BIC (smaller is better)
128.81
CAIC (smaller is better)
132.81
HQIC (smaller is better)
126.35
(b) Fit statistics for conditional distribution
-2 Log L (count | r. effects)
120.50
Pearson’s chi-square
9.21
Pearson’s chi-square/DF
0.58
(c) Fit statistics and Pearson’s chi-square/DF
Poisson
Negative binomial
175.35
120.50
-2 Log likelihood
AIC (smaller is better)
181.35
128.50
AICC (smaller is better)
183.35
132.13
BIC (smaller is better)
181.59
128.81
CAIC (smaller is better)
184.59
132.81
HQIC (smaller is better)
179.74
126.35
4.85
0.58
ϕ (Pearson’s chi-square/DF)
longer overdispersed ϕ = 0:58 . In other words, the negative binomial distribution
does a better job than the Poisson distribution in fitting these data, since it effectively
controls the overdispersion.
Comparing the fit statistics tabulated in Table 4.3 subsection (c) under both
distributions, we can observe that when the data are modeled under a negative
binomial distribution, the values of the fit statistics are lower than those under a
Poisson distribution, since the dispersion parameter ϕ < 1. This indicates that the
negative binomial models this dataset better.
4.8
4.8.1
Estimation and Inference in Generalized
Linear Mixed Models
Estimation
In GLMMs, inference involves the estimation and testing of the hypotheses of
unknown parameters in β, G, and R as well as the best linear unbiased predictions
(BLUPs) of random effects, b. In most modern statistical tools, including GLMMs,
parameter fitting is performed via maximum likelihood (ML) or methods derived
from this method. For simple analyses, in which the response variables are normal,
classical ANOVA methods are based on calculating the differences of the sums of
124
4 Generalized Linear Mixed Models for Non-normal Responses
the squares that produce the same results as an ML estimation. However, this
equivalence is not obtained in models with more complex structures such as
LMMs or GLMMs. To find the ML estimators, in GLMMs, one must integrate
over all possible values of the random effects. For GLMMs, this computation is at
best slow and at worst (a large number of random effects) computationally
infeasible.
Statisticians have proposed several ways to approximate the parameter estimates
of a GLMM, including penalized quasi-likelihood (PQL) and pseudo-likelihood
methods (Schall 1991; Wolfinger and O'Connell 1993; Breslow and Clayton
1993), Laplace approximations (Raudenbush et al. 2000) and Gauss–Hermite quadrature (Pinheiro and Chao (2006), and Bayesian methods based on Markov chain
Monte Carlo (Gilks et al. 1996). In all these approaches, researchers must distinguish
between a standard ML estimation, which estimates the standard deviations of the
random effects assuming that the fixed effects estimates are precisely correct, and
restricted maximum likelihood (REML), a variant that averages over the uncertainty
in the fixed effects parameters (Pinheiro and Bates 2000; Littell et al. 2006).
The ML method underestimates the standard deviations of random effects, except
in extremely large datasets, but it is most useful for comparing models with different
fixed effects. Pseudo- and quasi-likelihood methods are the simplest and the most
widely used in approximating a GLMM. They are widely implemented in statistical
packages that promote the use of GLMMs in many areas of ecology, biology, and
quantitative and evolutionary genetics (Breslow 2004). Unfortunately, pseudo- and
quasi-likelihood methods produce biases in parameter estimation if the standard
deviations of the random effects are large, especially when using binary data
(Rodriguez and Goldman 2001; Goldstein and Rasbash 1996). Lee and Nelder
(2001) have implemented several improvements to the PQL version, but these are
not available in most common statistical software packages. As a rule of thumb, PQL
performs poorly for Poisson data when the average number of counts per treatment
combination is less than five or for binomial data when the expected numbers of
successes and failures for each observation are less than five (Breslow 2004).
Another disadvantage of PQL is that it calculates a quasi-likelihood rather than the
true likelihood. Because of this, many statisticians believe that PQL-based methods
should not be used for inference.
There are two more accurate approximations available, which also reduce bias.
One is the Laplace approximation (Raudenbush et al. 2000), which approximates the
true likelihood of a GLMM instead of a quasi-likelihood, allowing the maximum
likelihood method in the GLMM inference process. The other approach is called
Gauss–Hermite quadrature (Pinheiro and Chao 2006), which is more accurate than
the Laplace approximation but is slower (requires more computational resources).
Therefore, the procedures for parameter estimation of a GLMM that are approximations are as follows:
The penalized quasi-likelihood method performs the estimation process by alternating between (1) estimating the fixed parameters by fitting a GLM with a variance–
covariance matrix based on an LMM fit and (2) estimating the variances and
4.8
Estimation and Inference in Generalized Linear Mixed Models
125
covariances by fitting a GLM with unequal variances calculated from the previous GLM fit. Pseudo-likelihood, a close cousin of the ML method, estimates
variances differently and estimates a scale parameter to account for
overdispersion (some authors use these terms interchangeably). In summary,
GLMMs require an iterative process in parameter estimation. Two categories of
iterative procedures are used by SAS: linearization and integral approximation.
The GLIMMIX procedure uses the pseudo-likelihood method in linearization,
and integral approximation uses the Laplace approximation or adaptive methods
such as Gauss–Hermite quadrature. These methods maximize the log likelihood
of the exponential distribution family, i.e., non-normal distributions. The pseudolikelihood method is the default procedure in the GLIMMIX procedure (Proc
GLIMMIX). The Laplace method and quadrature are an approximation for
maximum likelihood, but the Laplace method is computationally simpler than
quadrature and also provides excellent estimates.
4.8.2
Inference
After estimating the parameter values in a GLMM, the next step is to extract
information and draw statistical conclusions from a given dataset through careful
analysis of the parameter estimates (confidence intervals, hypothesis testing) and
select a model that best describes or explains the most variability in the dataset.
Inference can generally be based on three types: (a) hypothesis testing, (b) model
comparison, and (c) Bayesian approaches. Hypothesis testing compares test statistics
(F-test in ANOVA) to verify their expected distributions under the null hypothesis
(H0), estimating the value of P (P-value) to determine whether H0 can be rejected.
On the other hand, model selection compares candidate model fits. These can be
selected using hypothesis testing; that is, testing nested versus more complex models
(Stephens et al. 2005) or using information theory approaches such as Wald tests (Z,
χ 2, t, and F). In model selection, likelihood ratio (LR) tests can ensure the significance of factors or choose the best of a pair of candidate models. On the other hand,
information criteria allow multiple comparisons and selections of non-nested
models. Among these criteria are the Akaike information criterion (AIC) and related
information criteria that use deviance as a measure of fit, adding a term to penalize
more complex models. Information criteria can provide better estimates. Variations
of AIC are highly common when sample sizes are not large (AICC), when there is
overdispersion in the data (quasi-AIC, QAIC), or when one wishes to identify/
determine the number of parameters in a model (Bayesian information criterion,
BIC).
126
4
Generalized Linear Mixed Models for Non-normal Responses
4.9
Fitting the Model
The mathematics behind a GLMM is quite complex. It is difficult to conceptualize
the use of constructs such as distributions, link functions, log likelihood, and quasilikelihood when fitting a model. Perhaps the following points will help explain the
modeling process.
(a) An analysis of variance model is a vector of linear predictors (equation) with
unknown parameter estimates.
(b) Each distribution has a corresponding probability function.
(c) The vector of linear predictors is substituted into the likelihood function.
(d) Solutions to the parameter estimates are found by minimizing the negative of the
log likelihood function (-log likelihood).
(e) The means (least squares means – lsmeans) are derived from the parameter
estimates and are on the model scale.
(f) The link function converts the mean estimates at the model scale to the original
data scale.
The key concepts of proc GLIMMIX are (1) it uses a distribution to estimate the
model parameters; it does not fit the data to a distribution, and (2) the data values are
not transformed by the link function; the link function converts the means (least
squares means) to the data scale after estimation at the model scale.
4.10
Exercises
1. As a simple example of these types of data, consider the following results of an
experiment on wheat germination, carried out in pots under glass. The experiment
consisted of four blocks of six treatments (Table 4.6).
(a) According to the response variable, what type(s) of probability distribution do
you suggest for the variable?
(b) Construct a GLMM to study the effect of treatments on seed germination.
(c) Analyze the dataset according to the model proposed in (a). Is the probability
distribution proposed in (a) adequate?
(d) Is there a significant difference in the proportion of germinated seeds between
treatments?
Table 4.6 Number of seeds
not germinating (out of 50)
A
B
C
D
Trt1
10
8
5
1
Trt2
11
10
11
6
Trt3
8
3
2
4
Trt4
9
7
8
13
Trt5
7
9
10
7
Trt6
6
3
7
10
4.10
Exercises
127
Table 4.7 Control of cockchafer larvae
Trt1
Trt2
Trt3
Trt4
Trt5
A
a
3
4
3
5
4
B
7
3
10
8
6
B
a
7
3
6
4
4
b
15
12
12
11
11
C
a
5
4
4
1
2
b
7
2
4
5
2
D
a
5
12
1
5
3
b
14
5
14
9
8
E
a
0
2
2
2
0
b
3
3
2
7
1
F
a
1
1
1
3
5
b
7
6
7
7
4
G
a
1
3
1
0
1
b
10
5
8
3
6
H
a
4
4
7
3
1
b
13
11
10
12
8
2. Table 4.7 shows the counts per sample area of a variety type of cockchafer larva
(two age groups a and b). The experiment consisted of five treatments in eight
randomized blocks and two age groups to study the differential effects of
treatments on insect age.
(a) Considering the type of answer of this exercise; what type(s) of probability
distribution(s) do you suggest for this type of response?
(b) Construct a GLMM to study the effect of treatments and the age of Cockchafer larvae.
(c) Analyze the dataset according to the model proposed in (a).
(d) Is the model used in (a) sufficient? If so, discuss your findings.
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Chapter 5
Generalized Linear Mixed Models
for Counts
5.1
Introduction
Data in the for of counts regularly appear in studies in which the number of
occurrences is investigated, such as the number of insects, birds, or weeds in
agricultural or agroecological studies; the number of plants transformed or
regenerated using modern breeding techniques; the number of individuals with a
certain disease in a medical study; and the number of defective products in a quality
improvement study, among others. These counts can be counted per unit of time,
area, or volume. When using a generalized linear model (GLM) with a Poisson
distribution, it is often found that there is excessive dispersion (extra variation) that is
no longer captured by the Poisson model. In these cases, the data must be modeled
with a negative binomial distribution that has the same mean as the Poisson
distribution but with a variance greater than the mean. Most experiments have
some form of structure due to the experimental design (completely randomized
design (CRD), randomized complete block design (RCBD), incomplete block, or
split-plot design) or the sampling design, which must be incorporated into the
predictor to adequately model the data.
5.2
The Poisson Model
A Poisson distribution with parameter λ belongs to the exponential family and is a
discrete random variable, whose probability function is equal to
f ðyÞ =
e - λ λy
; λ > 0, y = 0, 1, 2, ⋯:
y!
© The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8_5
129
130
5
Generalized Linear Mixed Models for Counts
The mean and variance of a Poisson random variable are equal, i.e., E( y) = Var
( y) = λ. A Poisson distribution is often used to model responses that are “counts.” As
λ increases, the Poisson distribution becomes more symmetric and eventually it can
be reasonably approximated by a normal distribution.
Let yij be the value of the count variable associated with unit i at level one and
with unit j at level two, given a set of explanatory variables. Therefore, we can
express this as
y
f yij =
e - λij λijij
, yij = 0, 1, 2, ⋯
yij !
and the logarithm of the likelihood is given by:
y
log f yij = log
e - λij λijij
yij !
= - λij þ yij log λij - log yij ! :
A Poisson distribution has very particular mathematical properties that are used
when we model “counts.” For example, the expected value of y is equal to the
variance of y, such that
E ðyik Þ = Varðyik Þ = λij
Then, λij is necessarily a nonnegative number, which could lead to difficulties if
we consider using the identity bound function in this context. The natural logarithm
is mainly used as a link function for expected “counts.” For single-level (factor) data,
Poisson regression model is considered, where we work with the natural logarithm of
the counts, log(λi), whereas for multilevel data (more than two factors), mixed
models with Poisson data are considered a better choice for the logarithm of the
counts λij.
Suppose that given the random effects of b, the counts y1, y2, ⋯, yn are conditionally independent such that yij j bj~Poisson(λij), where
log λij = η þ τi þ bj :
This is a special case of a generalized linear mixed model (GLMM) in which the
link function of this family of distributions is g(λij) = log (λij). The dispersion
parameter ϕ, in this case, is equal to 1.
Sometimes, if the data counts are extremely large, their distribution can be
approximated to a continuous distribution. Whereas, if all the counts are large
enough, then the square root of the counts is viable for fitting the model as it allows
the variance to be stabilized. However, as mentioned in previous chapters, the
estimation process under normality can be problematic, as it can provide negative
fitted values and predictions, which is illogical.
5.2
The Poisson Model
5.2.1
131
CRD with a Poisson Response
An CRD is a design in which a fixed number of t treatments is randomly assigned to
r experimental units. The linear predictor describing the mean structure of this GLM
is
ηij = η þ τi
where ηij denotes the ijth link function of the ith treatment in the jth observation, η is
the intercept, and τi is the fixed effect due to treatment i (i = 1, 2, ⋯, t; j = 1, 2, ⋯ri),
with t treatments and ri replicates in each treatment i.
Example Effect of a subculture on the number of shoots during micropropagation
of sugarcane.
The objective of micropropagation in sugarcane is to produce vegetative material
identical to the donor so that its genetic integrity is preserved. Despite this,
somaclonal variation has been observed in plants derived from in vitro culture
regardless of explant, variety, ploidy level, number of subcultures, and generation
route used, among others. A total of 8 explants were planted in temporary immersion
bioreactors (explant/bioreactor) to determine whether the number of subcultures
(10 subcultures) influences the number of shoots observed per explant. In this
example, we have ri observations ( j = 1, 2, . . ., ri) on each of the 10 subcultures
(i = 1, 2, . . ., 10) in a completely randomized design (Appendix 1: Data: Subcultures). The analysis of variance (ANOVA) table (Table 5.1) for this model is given
below:
The components of the GLM are set out below:
Distribution: yij Poisson λij
Linear predictor: ηij = η þ τi
Link function: log λij = ηij
where yij denotes the number of sprouts observed in subculture i explant j (i = 1, 2,
⋯, 10; j = 1, 2, ⋯, 8), ηij is the ijth link function, η is the intercept, and τi is the fixed
effect of subculture i.
Table 5.1 Analysis of variance
Sources of variation
Subculture
Error
Total
Degrees of freedom
Unbalanced design
t - 1 = 10 - 1 = 9
10
i = 1 ri
10
i = 1 ri
- t = 164
Balanced design
t-1
t(r - 1)
- 1 = 173
tr - 1
132
Table 5.2 Model information
and estimation methods
5
Generalized Linear Mixed Models for Counts
(a) Model information
Dataset
Response variable
Response distribution
Link function
Variance function
Variance matrix
Estimation technique
Degrees of freedom method
(b) Dimensions
Columns in X
Columns in Z
Subjects (blocks in V)
Max Obs per subject
WORK.SUGAR
NB
Poisson
Log
Default
Diagonal
Maximum likelihood
Residual
0
1
The following Statistical Analysis Software (SAS) code allows analyzing an
CRD with a Poisson response.
proc glimmix data=sugar method=laplace;
class rep1 sub1 ;
model nb=sub/dist=poisson s link=log;
lsmeans sub/lines ilink;
run;quit;
While most of the commands used have been explained before, the options in the
model statement “dist,” “s,” and “link” communicate to the SAS the type of data
distribution, the fixed effects solution, and the link to use, respectively. In addition,
the “lines” option asks the GLIMMIX procedure in the “lsmeans” (least squares
means) command for mean comparisons, and the “ilink” option provides the inverse
link function.
Part of the output is shown in Table 5.2, where part (a) shows the model and the
methods used to fit the statistical model, whereas part (b) lists the dimensions of the
relevant matrices in the model specification.
Due to the absence of random effects in this model, there are no columns in
matrix Z. The 11 columns in matrix X comprise an intercept and 10 columns for the
effect of subcultures.
The goodness-of-fit statistics of the model are shown in part (a) of Table 5.3. The
value of the generalized chi-squared statistic over its degrees of freedom (DFs) is
less than 1. (Pearson′s chi - square/DF = 0.79). This indicates that there is no
overdispersion and that the variability in the data has been adequately modeled with
the Poisson distribution.
Subsection (b) of Table 5.3 shows the maximum likelihood (ML) (“Estimate”),
parameter estimates, standard errors, and t-tests for the hypothesis of the parameters.
5.2
The Poisson Model
133
Table 5.3 Fit statistics and estimated parameters
(a) Fit statistics (Akaike’s information criterion (AIC), a small sample bias Corrected Akaike’s
information criterion (AICC), Bozdogan Akaike’s information criteria (CAIC), Schwarz’s
Bayesian information criterion (BIC), Hannan and Quinn information criterion (HQIC))
-2 Log likelihood
1062.11
Akaike information criterion (AIC) (smaller is better)
1082.11
AICC (smaller is better)
1083.46
Bayesian information criterion (BIC) (smaller is better)
1113.70
CAIC (smaller is better)
1123.70
HQIC (smaller is better)
1094.93
Pearson’s chi-square
137.70
Pearson’s chi-square/DF
0.79
(b) Parameter estimates
Effect
sub1
Estimate
Standard error
DF
t-value
Pr > |t|
^η
Intercept
3.6687
0.04124
164
88.96
<0.0001
-1.0809
0.07389
164
-14.63
<0.0001
sub1
1
^τ1
sub1
2
^τ2
-0.9043
0.06664
164
-13.57
<0.0001
-0.5596
0.06839
164
-8.18
<0.0001
sub1
3
^τ3
-0.3412
0.06398
164
-5.33
<0.0001
sub1
4
^τ4
0.2177
0.05540
164
3.93
0.0001
sub1
5
^τ5
sub1
6
^τ6
0.2257
0.05452
164
4.14
<0.0001
0.2631
0.05178
164
5.08
<0.0001
sub1
7
^τ7
0.3387
0.05109
164
6.63
<0.0001
sub1
8
^τ7
0.2684
0.05478
164
4.90
<0.0001
sub1
9
^τ8
sub1
10
^τ10
0
.
.
.
.
Table 5.4 (part (a)) shows significance tests for the fixed effects in the model
“Type III fixed effects tests.” These tests are Wald tests and not likelihood ratio tests.
The effect of a subculture on the number of shoots is highly significant in this model
with a value of P < 0.0001, indicating that the 10 subcultures do not produce the
same number of shoots, that is, the number of subcultures affects the average shoot
production in the explant.
The least squares means obtained with “lsmeans” (part (b) in Table 5.4) are the
values under the column “Estimate,” which along with the standard errors, were
calculated with the linear predictor ηi = η þ τi . These estimates are on the model
scale, whereas the “Mean” column values and their respective standard errors are on
the data scale, which were obtained by applying the inverse link to obtain the λi
values, i.e., λi = exp ðηi Þ with their respective standard errors.
A comparison of means, using the option “lines,” is presented in Fig. 5.1. In this
figure, we can see that in the first subcultures, the average production is minimal but
it increases as subcultures increase from 5 to 8, and, in subculture 9, the average
number of shoots per explant begins to decrease.
134
5
Generalized Linear Mixed Models for Counts
Table 5.4 Type III tests of fixed effects and least squares means (means)
(a) Type III tests of fixed effects
Effect
Num DF
sub1
9
(b) sub1 least squares means
sub1 Estimate Standard error
1
2.5878
0.06131
2
2.7644
0.05234
3
3.1091
0.05455
4
3.3274
0.04891
5
3.8864
0.03699
6
3.8944
0.03567
7
3.9318
0.03131
8
4.0073
0.03015
9
3.9370
0.03606
10
3.6687
0.04124
errorstd ðηi Þ
ηi
Den DF
164
DF
164
164
164
164
164
164
164
164
164
164
t-value
42.21
52.81
56.99
68.03
105.08
109.18
125.57
132.91
109.18
88.96
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
13.3000
15.8696
22.4000
27.8667
48.7333
49.1250
51.0000
55.0000
51.2667
39.2000
Standard error mean
0.8155
0.8307
1.2220
1.3630
1.8025
1.7522
1.5969
1.6583
1.8487
1.6166
λi
errorstd λi
60
Average of shoots per explant
Pr > F
<0.0001
F-value
120.14
a
b
50
b
ab
ab
c
40
d
30
20
e
g
f
10
0
1
2
3
4
5
6
7
8
9
10
Subcultures
Fig. 5.1 Average number of shoots per subculture. Bars with different letters are statistically
different using α = 0.05
5.2.2
Example 2: CRDs with Poisson Response
Researchers want to determine whether the application of a new growth compound
to walnut trees changes the amount of nuts produced per tree. They were applied at
three different times (pre-flowering = 1, flowering = 2, and post-flowering = 3) and
5.2 The Poisson Model
Table 5.5 Number of nuts
per tree (yij) in each of the
combinations of the two
factors
135
Trt
C
A2
A1
B1
B1
B2
A1
yij
1
118
69
89
99
158
89
Trt
A3
B3
A1
B3
A3
C
A2
yij
79
99
50
118
99
50
127
Trt
A3
C
A2
B3
A1
B1
A2
yij
89
21
79
99
79
118
89
Trt
B1
A2
A3
A1
B2
C
B2
yij
50
69
69
21
118
30
99
Trt
B2
B1
A3
C
B2
B3
B3
yij
138
69
138
11
89
158
118
in two formulations (A and B) plus a control (C). In addition to the treatments (Trt)
there was a control, where no compound was applied. In total, 7 treatments were
randomly applied to the experimental units (trees), i.e., 35 trees, in a rectangular
arrangement (as shown below). The average number of nuts yij observed in the
formulation and the time of application are provided in Table 5.5.
The components of the GLMM are listed below:
Distribution : yij j r j Poissonðλij Þ
r j Nð0, σ2tree Þ
Linear predictor: ηij = η þ τi þ r j
Link function: log λij = ηij
where yij denotes the number of nuts in treatment i on tree j (i = 1, 2, ⋯, 7; j = 1, 2,
⋯, 5), ηij is the linear predictor, η is the intercept, τi is the fixed effect due to
treatment i, and rj is the random effect due to tree j.
The following SAS statements allows a GLMM to be fitted in a completely
randomized design with a Poisson response variable.
proc glimmix data=crd_nuez nobound method=laplace;
class trt rep;
model count = trt/dist=Poi link=log;
random rep;
lsmeans trt/lines ilink;
run;
The options in the model statement, dist, s and ilink communicates to SAS the
type of data distribution, the fixed effects solution and to compute the inverse link,
respectively. In addition, the option “lines” requests the GLIMMIX procedure in the
“lsmeans” (least squares means) command, and the mean comparisons and the
“ilink” option provide the inverse of the link function.
Part of the results is presented in Table 5.6. The value of the statistic for
conditional distribution (part (a)) indicates that there is a strong overdispersion (χ 2/
df = 3.62), and the variance component estimates due to sampling in the experimental units (trees) is σ 2tree = 0:035 (part (b)).
136
5
Table 5.6 Results of the
analysis of variance
Generalized Linear Mixed Models for Counts
(a) Fit statistics for conditional distribution
-2 Log L (count | r. effects)
354.60
Pearson’s chi-square
126.54
Pearson’s chi-square/DF
3.62
(b) Covariance parameter estimates
Cov Parm
Estimate
Standard error
rep
0.03573
0.02362
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Trt
6
24
59.55
<0.0001
In addition, Table 5.6 (part (c)) shows the type III tests of fixed effects, indicating
that there is a significant difference between treatments on the average number of
nuts per tree (P = 0.0001). However, it is not recommended to continue with the
inference and analysis of the experiment due to the presence of extra-variance
(commonly known as overdispersion; Pearson′s chi - square/DF = 3.62) in the
data that strongly affects the F-test and the standard errors of the means.
A highly effective alternative to deal with the inconvenience of overdispersion in
the data is to use a different distribution to the Poisson distribution. A negative
binomial distribution is an excellent option for count data with overdispersion.
Assuming that the conditional distribution of the observations is given by:
yij j r j Poisson λij ,
where λij~Gamma~(1/ϕ, ϕ), ϕ as the scale parameter and r j Nð0, σ2tree Þ. The
resulting new GLMM is:
Distribution : yij j r j Negative Binomialðλij , ϕÞ,
r j Nð0, σ2tree Þ
Linear predictor : ηij = η þ τi þ r j
Link function: log λij = ηij
The following GLIMMIX statements for fitting this model under a negative
binomial distribution in a CRD manner is provided next.
proc glimmix data=crd_nuez nobound method=laplace;
class trt rep;
model count = trt/dist=Negbin link=log;
random rep;
lsmeans trt/lines ilink;
run;
5.2
The Poisson Model
Table 5.7 Poisson and negative binomial model fit
statistics
Table 5.8 Variance component estimates and fixed
effects tests
137
(a) Fit statistics
Poisson
Negative
-2 Log likelihood
374.83
AIC (smaller is better)
390.83
AICC (smaller is better)
396.37
BIC (smaller is better)
387.71
CAIC (smaller is better)
395.71
HQIC (smaller is better)
382.45
(b) Fit statistics for conditional distribution
Poisson
Negative
-2 Log L (count | r. effects)
354.60
Pearson’s chi-square
126.54
Pearson’s chi-square/DF
3.62
(a) Covariance parameter estimates
Cov Parm
Estimate
Rep
0.04288
Scale
0.06141
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
Trt
6
24
Binomial
328.03
346.03
353.23
342.51
351.51
336.60
Binomial
316.06
32.02
0.91
Standard error
0.03398
0.02428
F-value
18.75
Pr > F
<0.0001
Part of the results is listed below. The information criteria in Table 5.7 part (a) are
helpful in choosing which model best fits the dataset. Clearly, the negative binomial
distribution provides the best fit to these data. On the other hand, in the conditional fit
statistics (part (b)), we observed that the Poisson model had a strong overdispersion
(Pearson′s chi - square/DF = 3.62) and that by fitting the data under a negative
binomial distribution, the overdispersion of the dataset was removed (Pearson′s chi Square/DF = 0.91).
Table 5.8 shows the variance component estimates (part (a)) and the type III tests
of fixed effects (part (b)). The estimated variance parameter, due to trees, is
σ 2tree = 0:04288, and the estimated scale parameter (Scale) is ϕ = 0:06141. The
type III tests of fixed effects (part (b)) show that there is a highly significant effect
of treatments on the average number of nuts (P < 0.0001).
The values under the column “Estimates” are the estimates of the linear predictor
ηi (the model scale), and the values under “Mean” are the means λi (the data scale)
with their respective standard errors obtained with the command “lsmeans” and
“ilink” (Table 5.9). The results show that the treatments implemented in this
experiment showed a higher average number of walnuts than did the “control”
treatment C. In general, formula B applied to the walnut trees at the full-flowering
stage showed a higher nut production.
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Generalized Linear Mixed Models for Counts
Table 5.9 Estimates on the model scale (“Estimate”) and means on the data scale (“Mean”)
Trt least squares means
Trt Estimate Standard error
A1 4.0865
0.1560
A2 4.5624
0.1519
A3 4.5293
0.1519
B1 4.4349
0.1529
B2 4.7863
0.1504
B3 4.7641
0.1504
C
3.0499
0.1742
DF
24
24
24
24
24
24
24
t-value
26.19
30.04
29.82
29.01
31.82
31.67
17.51
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
59.5307
95.8162
92.6956
84.3417
119.86
117.23
21.1140
Standard error mean
9.2890
14.5529
14.0783
12.8958
18.0304
17.6335
3.6785
Interest often arises in areas of agricultural and biological sciences to conduct
experiments that involve random effects (blocks, locations, etc.) and response variables different from the normal distribution. For example, suppose that a certain
number of treatments are being tested at different randomly selected locations, out of
a sufficiently large number of locations. At each location, the experimental units are
randomly assigned to each of the treatments. Let yij be the number of (observed)
individuals possessing the characteristic of interest in the ith treatment in the jth
block. The model for the mean structure of this experiment is
ηij = η þ τi þ bj
where η is the intercept, τi is the fixed effect due to the ith treatment i, and bj is the
random effect of the block j with bj Nð0, σ2block Þ.
5.2.3
Example 3: Control of Weeds in Cereal Crops
in an RCBD
One of the main problems when growing cereal crops is the competition that exists
between the weeds and seedlings. If a field supervisor is interested in testing five
designed treatments plus a control for weed control in cereal crops, then a randomized complete block design (four blocks) should be used. Table 5.10 shows the
number of weed plants observed in each of the treatments (yij) in parentheses.
Table 5.11 shows the sources of variation and the degrees of freedom of a
randomized complete block design used in this experiment.
Since the response is count, it will be modeled using a GLMM with a Poisson
response variable, which is stated below:
Distribution : yij j bj Poissonðλij Þ
bj Nð0, σ2block Þ
5.2
The Poisson Model
139
Table 5.10 Number of weeds in each treatment (the number in parentheses corresponds to the
treatment number)
Block
A
B
C
D
(1) 438
(3) 61
(5) 77
(2) 315
Table 5.11 Analysis of
variance
(4) 17
(2) 422
(3) 157
(1) 380
(2) 538
(6) 57
(4) 87
(5) 20
(5) 18
(1) 442
(6) 100
(3) 52
Sources of variation
Block
Treatment
Error
Total
(3) 77
(5) 26
(2) 377
(4) 16
(6) 115
(4) 31
(1) 319
(6) 45
Degrees of freedom
b-1=4-1=3
t-1=6-1=5
(t - 1(b - 1)=15
tb - 1 = 23
Linear predictor: ηij = η þ τi þ bj
Link function: log λij = ηij
where yij denotes the number of weed plants observed in treatment i and block
j (i = 1, 2, ⋯, 6; j = 1, 2, 3, 4), ηij is the linear predictor, η is the intercept, τi is the
fixed effect due to treatment i, and bj is the random block effect bj N 0, σ 2block .
Using the GLIMMIX procedure, the following syntax specifies the analysis of a
GLMM with a Poisson response.
proc glimmix nobound method=laplace;
class Block Trt;
model Count = Trt/dist=Poisson s;
random block;
lsmeans Trt/diff lines ilink;
run; quit;
Note that in the above syntax, we use “method = laplace” (or we can also use
“method = quadrature”) to fit the mixed model and obtain the chi-squared/DF fit
statistic. If the method of integration is not specified, then a generalized chi-squared/
DF statistic is obtained. The auxiliary options after the “lsmeans” command are
described below: “diff” provides paired comparisons between treatments, “lines”
provides the pair comparison of means using letters, and “ilink” provides the value
of the inverse of the link function. Some of the outputs are listed below.
Table 5.12 (a) presents the basic information about the model and estimation
procedure used.
Subsection (b) of Table 5.12 shows/ lists the “Dimensions” of the relevant
matrices used in the model. The random effects matrix Z indicates that there are
four columns due to blocks, and the fixed effects matrix X indicates that there is one
column for the intercept plus six columns due to treatments.
140
Table 5.12 Basic model
information
Table 5.13 Model fit
statistics
5
Generalized Linear Mixed Models for Counts
(a) Model information
Dataset
Response variable
Response distribution
Link function
Variance function
Variance matrix
Estimation technique
Likelihood approximation
Degrees of freedom method
(b) Dimensions
G-side Cov. parameters
Columns in X
Columns in Z
Subjects (blocks in V)
Max Obs per subject
WORK.DBCA
Counting
Poisson
Log
Default
Not blocked
Maximum likelihood
Laplace
Containment
(a) Fit statistics
-2 Log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
(b) Fit statistics for conditional distribution
-2 Log L (Count | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF ϕ
1
7
4
1
24
434.46
448.46
455.46
444.16
451.16
439.03
418.66
283.09
11.80
The “Fit statistics” and “Fit statistics for conditional distribution” (parts (a) and
(b) of Table 5.13, respectively) show information about the fit of the GLMM. The
generalized chi-squared statistic measures the sum of the residual squares in the final
model and the relationship with its degrees of freedom; this is a measure of the
variability of the observations about the model around the mean.
The value of Pearson’s chi-square/DF for the conditional distribution is 11.8, well
above up 1. This value gives strong evidence of overdispersion in the dataset. In
other words, this value is calling our distribution and linear predictor assumption into
question, which means that the variance function was not adequately specified.
5.2
The Poisson Model
141
Table 5.14 Variance component estimates, parameter estimates, and type III tests of fixed effects
Cov Parm
ð^
σ2block Þ
Block
(a) Solutions for fixed effects
Effect
Trt
Estimate
η
4.3637
Intercept
1.6056
Trt
1
τ1
τ2
1.6508
Trt
2
τ3
0.09042
Trt
3
Trt
4
τ4
-0.7416
τ5
-0.8101
Trt
5
τ6
0
Trt
6
(b) Type III tests of fixed effects
Effect
Num DF
Trt
5
Estimate
0.01840
Standard error
0.01377
Standard error
0.08808
0.06155
0.06132
0.07769
0.09888
0.1012
.
DF
3
15
15
15
15
15
.
Den DF
15
t-value
49.54
26.09
26.92
1.16
-7.50
-8.00
.
Pr > |t|
<0.0001
<0.0001
<0.0001
0.2627
<0.0001
<0.0001
.
Pr > F
<0.0001
F-value
523.57
Table 5.15 Estimated least squares means (“Mean”)
Trt least squares means
Trt Estimate Standard error
1
5.9693
0.07237
2
6.0145
0.07217
3
4.4541
0.08652
4
3.6221
0.1060
5
3.5536
0.1081
6
4.3637
0.08808
errorstd ðηi Þ
ηi
DF
15
15
15
15
15
15
t-value
82.49
83.33
51.48
34.19
32.86
49.54
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
391.25
409.34
85.9802
37.4150
34.9372
78.5467
Standard error mean
28.3139
29.5437
7.4390
3.9643
3.7784
6.9186
λi
errorstd λi
The F-test for testing H0 (τ1 = τ2 = ⋯ = τ6) or equivalent (μ1 = μ2 = ⋯ = μ6)
indicates that there is a highly significant difference (P < 0.0001) in the average
number of weeds in at least one treatment (part (c)) (Table 5.14).
The estimates of the linear predictor on the model scale for each of the treatments
ðηi Þ and the inverse of the linear predictor λi on the data scale (with their
respective standard errors) are calculated as follows ηi = η þ τi and λi = exp ðηi Þ ,
respectively. These values are listed in Table 5.15.
The “plots” option in the “proc GLIMMIX” statement creates a set of plots for the
raw residuals, Pearson residuals, and studentized residuals.
The panel consists of a plot of studentized residuals versus the linear predictor
ðηi Þ, a histogram of the residuals with a normal density superimposed, a plot of
residual versus quantiles, and a box plot for the residuals. The panel of studentized
residuals indicates the possibility of a slightly skewed distribution (Fig. 5.2). In this
figure, we can see that the range of values of the residuals changes, as do the values
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Generalized Linear Mixed Models for Counts
Fig. 5.2 Studentized conditional residuals
of the linear predictor, indicating that the assumption of constant variance is no
longer met. The residuals–quantiles plot confirms the constant variance violation. A
nonconstant variance may also suggest an incorrect selection of the response distribution or variance function.
5.2.4
Overdispersion in Poisson Data
Linear mixed models assume that the observations have a normal distribution
conditional to the fixed effects of parameters. In addition, the mean μ is independent
of the variance σ 2, whereas, in most GLMMs that assume a binomial or Poisson
distribution, the variance “dispersion” is set to 1. That is, if the mean is known, then
we assume that the variance is also known. The extra variability not predicted
by a generalized linear model’s random component reflects overdispersion.
Overdispersion occurs because the mean and variance components of a GLM are
related and depend on the same parameter that is being predicted through the
predictor set. However, if overdispersion is present in a dataset, then the estimated
standard errors and test statistics of the overall goodness of fit will be distorted and
5.2
The Poisson Model
143
adjustments must be made. In other words, when there is overdispersion in a dataset,
the standard errors of the estimated parameters are too small, which leads to test
statistics for the model parameters that are too large (i.e., type I error increases).
Overdispersion can be caused by several factors: omission of predictor variables
in the model, high correlation in the observations due to nested effects,
misspecification of the systematic component, or incorrect distribution of the data.
Systematic or overdispersion deviations may be the result of incorrect assumptions
about the stochastic and/or systematic component of the model. The model may also
not fit the dataset well because of an incorrect choice of the link function. Systematic
deviations may also result from lack of either random effects or independence of
observations. These random factors should generally address deviance violations
and problems associated with the systematic component.
According to Stroup (2013), overdispersion occurs when the variance exceeds the
theoretical variance under the distribution model of the data. For any distribution
with a nontrivial variance function, overdispersion is theoretically possible for
distributions belonging to the one-parameter exponential family because they lack
a scale parameter to mitigate the mean–variance relationship; therefore, models such
as Poisson distribution are vulnerable to overdispersion. In summary, overdispersion
occurs when:
(a) The variance is larger than expected, which leads to standard errors that are not
correct.
(b) The mean structure is not well specified.
(c) The linear predictor η is not well specified.
(d) The chosen distribution of the data is not appropriate.
(e) Predictor variables are omitted.
(f) Observations are significantly correlated.
If we do not account for overdispersion, we underestimate the standard errors (for
a large variance, the standard errors are not correct) and inflate the statistical tests
causing the type I error to inflate and the confidence intervals to be unreliable.
Fig. 5.3 shows that as the predicted mean μ increases, the residuals have a larger
spread in the plot, indicating that the variance may increase as a function of the
mean, whereas Fig. 5.4 shows a nonconstant variance.
In the fit statistics obtained under the GLMM with the Poisson distribution (part
(b), Table 5.13), the value of the statistic of Pearson′s chi - square/DF = 11.8)
indicates that there is a strong overdispersion in the dataset. Another aspect provided
by the output is the value of the test statistic F (F = 523.57) tabulated in (part (c)) of
Table 5.14. A value too large may indicate that the fit is incorrect. Once the
researcher has detected overdispersion, he/she must consider the strategy that will
take to remedy it. There are three possible alternatives to evaluate (test) and eliminate
overdispersion. Below, we will review the three aforementioned alternatives.
144
5
Generalized Linear Mixed Models for Counts
Fig. 5.3 Conditional residuals versus predicted values on the data scale
5.2.4.1
Using the Scale Parameter
The first alternative is to add a scale parameter and replace Var(yij| bj) = λij by
Var(yij| bj) = ϕλij. This consists of replacing the logarithm of the conditional
likelihood yij log (λij) - λij - log (yij) by the quasi-likelihood yij log (λij) - λij/ϕ,
assuming that ϕ > 1 could adequately model the observed variance.
The following GLIMMIX syntax invokes this alternative of adding a scale
parameter under a Poisson response variable.
proc glimmix;
class Block Trt;
model Count = Trt/dist=Poisson;
random intercept/subject=block;
random _residual_;
lsmeans Trt/ ilink ;
run;
The SAS code is highly similar to that previously used with the addition of the
“random _residual_” command to the program. Note that the Laplace integration
method (“method = laplace”) has been removed, which causes the estimation to be
performed using the pseudo-likelihood (PL) method; the scale parameter is estimated and used in the adjustment of the standard errors and test statistics. The
5.2
The Poisson Model
145
Fig. 5.4 Residuals on the model scale
GLIMMIX procedure uses the generalized chi-square divided by its degrees of
freedom Gener:chi - square=DF = ϕ as the estimate of the scale parameter. All
standard errors are multiplied by ϕ, and all F-test values are divided by ϕ.
Table 5.16 shows part of the results.
In Table 5.16, we observe the fit statistics (part (a)), covariance parameter
estimates (part (b)), and the value of the scale parameter, which is equal to
ϕ = 19:4848 (Residual(VC)). The value of the F-statistic under the Poisson distribution in the analysis is 26.87 (part (c)); this value is obtained by dividing the
F-value from the previous analysis by 523:57=ϕ . The results indicate that even
under this adjustment, overdispersion exists and that this value increases from 11.8
to 19.4848 (part (a)). The inclusion of the scale parameter affects the variance
estimate due to blocks σ 2block as well as the estimates of treatment means (part (d)),
but the main impact is on the standard errors.
The inclusion of the scale parameter implies that there is a quasi-likelihood,
meaning that there is no true likelihood of the model and, therefore, there is no
true likelihood process that provides a true expected value of λ and a variance of ϕλ.
146
5
Generalized Linear Mixed Models for Counts
Table 5.16 Results of the adjustment by adding the scale parameter
(a) Fit statistics
-2 Res log pseudo-likelihood
Generalized chi-square
Gener. chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Intercept
Residual Variance component (VC)
(c) Type III tests of fixed effects
Effect
Num DF
Trt
5
(d) Trt least squares means
Trt Estimate Standard error DF
1
5.9779
0.1166
15
2
6.0231
0.1142
15
3
4.4626
0.2396
15
4
3.6306
0.3609
15
5
3.5621
0.3734
15
6
4.3722
0.2504
15
5.2.4.2
29.48
350.73
19.48
Subject
block
Estimate
0.004981
19.4848
Den DF
15
t-value
51.29
52.74
18.63
10.06
9.54
17.46
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
F-value
26.87
Mean
394.60
412.84
86.7160
37.7352
35.2362
79.2189
Standard error
0.02077
7.1346
Pr > F
<0.0001
Standard error mean
45.9935
47.1443
20.7753
13.6205
13.1576
19.8383
Linear Predictor Review
In count and binomial response variables, it is important to check whether the linear
predictor is correctly specified, that is, whether it is being randomly affected by the
experimental units within blocks. If λij is being randomly affected by the experimental units within blocks, which is important in count and binomial response
variables, then, the ANOVA table should include the effect of the block × treatment
source of variation; this must be specified in the linear predictor in a GLMM. Thus,
the linear predictor is specified as
ηij = η þ τi þ bj þ ðbτÞij
Distribution : yij j bj , bτij Poissonðλij Þ
bj Nð0, σ2block Þ
bτij Nð0, σ2block × τ Þ
Linear predictor: ηij = η þ τi þ bj þ ðbτÞij
Link function: log λij = ηij :
The following GLIMMIX program allows the above model to be adjusted:
5.2
The Poisson Model
147
Table 5.17 Results of the fit by redefining the predictor of the model
(a) Fit statistics for conditional distribution
-2 Log L (Count | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Subject
Estimate
Intercept
block
0.05969
Trt
block
0.1152
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Trt
5
15
41.48
(d) Trt least squares means
Trt Estimate Standard error DF t-value Pr > |t|
Mean
1
5.9692
0.2106
15
28.34
<0.0001 391.20
2
6.0037
0.2106
15
28.51
<0.0001 404.92
3
4.3674
0.2170
15
20.12
<0.0001
78.8402
4
3.4255
0.2301
15
14.89
<0.0001
30.7370
5
3.4005
0.2298
15
14.80
<0.0001
29.9786
6
4.3027
0.2175
15
19.79
<0.0001
73.8997
156.38
2.58
0.11
Standard error
0.05758
0.04115
Pr > F
<0.0001
Standard error mean
82.3947
85.2707
17.1100
7.0714
6.8891
16.0707
proc glimmix method=laplace;
class Block Trt;
model Count = Trt/dist=Poisson;
random intercept Trt/subject=block;
lsmeans Trt/ ilink ;
run;
Part of the output is shown in Table 5.17. The results tabulated in part (a) indicate
that the overdispersion has been eliminated ϕ = 0:11 , but there is a risk of
underestimating the variance. For this reason, it is highly recommended that the
value of ϕ should be close to 1. The estimated variance components (part (b)) for
blocks and block × treatments are σ 2block = 0:05969 and σ 2block × Trt = 0:1152,
respectively.
The type III tests of fixed effects are highly significant (P = 0.0001), indicating
that the six treatments are not equally effective in weed control (part (c)). The values
in part (d) under the “Mean” column are the means on the original scale of the data
for each of the treatments with their respective standard errors. The values of the
means – compared with the previous ones – (using the scale parameter) do not vary
much, but the standard errors have a more marked variation.
148
5.2.4.3
5
Generalized Linear Mixed Models for Counts
Using a Different Distribution
Another way to account for the problem of overdispersion when using a Poisson
distribution is to change the assumed distribution of the response variable. Poisson
variables have the same mean and variance, but, in biological sciences, with variables such as counts, this assumption is not always true. A negative binomial
distribution is a good alternative (see Example 5.2), as previously discussed. A
negative binomial variable’s mean is denoted by the parameter λ > 0 and variance
λ + ϕλ2 by ϕ > 0. That is, the expected value E( y) = λ and variance Var( y) = λ + ϕλ2,
where ϕ is the scale parameter. The components of this model are shown below:
Given that yij j bj~Poisson(λij), it is assumed that λij~Gamma~(1/ϕ, ϕ), with ϕ as
the scale parameter and bj N 0, σ 2block . The new specification of the resulting
GLMM is as follows:
Distribution : yij j bj Negative Binomialðλij , ϕÞ
bj Nð0, σ2block Þ
Linear predictor: ηij = η þ τi þ bj
Link function: log λij = ηij :
The following GLIMMIX statements fit the model with a negative binomial
distribution.
proc glimmix method=laplace;
class block Trt;
model count = Trt/dist=NegBin;
random block;
lsmeans Trt/ ilink ;
run;
Some of the most relevant outputs from GLIMMIX are presented in Table 5.18.
Pearson’s chi-squared (Pearson′s chi - square/DF) value of 0.88 (part (a)) shows
that overdispersion in the dataset has been removed. The estimated scale parameter
tabulated in part (b) (Scale) is ϕ = 0:1080. This value is not the same scale parameter
estimated using the Poisson model with the “random _residual_” command, since
the methodology for calculating them in these models is different. However, as
mentioned above, both scale parameters affect the relationship between the mean
and variance in the Poisson and negative binomial distributions.
The value of the test statistic shown in part (c) of Table 5.18, under the negative
binomial distribution for the effect of treatments, is highly similar to the value
obtained with the Poisson distribution when the effect of the block × treatment
interaction was added to the linear predictor. The values under “Estimate” are
estimates of the linear predictor on the model scale (part (d)), whereas those under
the “Mean” column are the treatment means on the data scale, using the negative
binomial distribution. Of the three proposed alternatives to fit these data, the last two
(including in the predictor the block–treatment interaction and assuming a negative
binomial distribution) provides a better fit.
5.2
The Poisson Model
149
Table 5.18 Fitting results by redefining the model structure
(a) Fit statistics for conditional distribution
-2 Log L (Count | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Estimate
Block
0.07713
Scale
0.1080
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
Trt
5
15
(d) Trt least squares means
Trt Estimate Standard error DF t-value Pr > |t|
1
6.0280
0.2179
15
27.66
<0.0001
2
6.0465
0.2174
15
27.81
<0.0001
3
4.3941
0.2228
15
19.72
<0.0001
4
3.5190
0.2335
15
15.07
<0.0001
5
3.4684
0.2338
15
14.83
<0.0001
6
4.3439
0.2235
15
19.44
<0.0001
5.2.5
235.11
21.13
0.88
Standard error
0.06955
0.03768
F-value
41.11
Mean
414.90
422.64
80.9704
33.7516
32.0863
77.0085
Pr > F
<0.0001
Standard error mean
90.4085
91.8789
18.0426
7.8815
7.5030
17.2111
Factorial Designs
Many experiments involve studying the effects of two or more factors. Factorial
designs are the most efficient for these types of experiments. In a factorial design, all
possible combinations of factor levels are investigated in each replicate. If there are
a levels of factor A and b levels of factor B, then each replicate contains all ab
treatment combinations.
5.2.5.1
Example: A 2 × 4 Factorial with a Poisson Response
This application refers to a factorial experiment involving explants from cotyledons
of cucumber (Cucumis sativus L.) with two factors, i.e., genotype (two levels) and
culture medium (four levels). Each of the eight combinations of the genotype and
culture levels were applied to four Petri dishes, each containing six leaf explants. The
response variable was the number of buds in each of the leaf explants, i.e., the
response variable was a count. There are two sources of variation in this application,
namely, variation between Petri dishes and variation between the explants within the
Petri dishes (Table 5.19).
The sources of variation and degrees of freedom for this experiment are shown in
Table 5.20.
150
5
Generalized Linear Mixed Models for Counts
The components that define this model are shown below:
Distribution : yijk j petri:dishk ,
explanteðpetri:dishÞlðkÞ Poissonðλijk Þpetri:dishj N 0, σ2petri:dish ,
explantðpetri:dishÞlðkÞ N 0, σ2explantðpetri:dishÞ
Linear predictor : ηijkl = η þ αi þ βj þ ðαβÞij þ petri:dishk þ explantðpetri:dishÞlðkÞ
Link function: log λijkl = ηijkl
where ηijkl is the linear predictor in genotype i (i = 1, 2), culture medium
j ( j = 1, 2, 3, 4), Petri.dish k (k = 1, 2, 3, 4), and explant l (l = 1, 2, 3, 4, 5, 6), η
is the intercept, αi is fixed effect due to genotype i, βj is the fixed effect due to
culture medium j, (αβ)ij is the effect of the interaction between genotype i and
culture medium j, Petri.dishk is the random effect of the Petri.dish, and explant
(Petri.dish)l(k) is the random effect of the explant within the Petri.dish, assuming
Petri:dishj Nð0, σ2Petri:dish Þ and explantðPetri:dishÞlðkÞ Nð0, σ2explantðPetri:dishÞ Þ.
The following GLIMMIX procedure fits a factorial experiment with a Poisson
response.
proc glimmix method=laplace ;
class genotype culture petri.dish explant;
model y = genotype|culture/dist=Poisson;
random petri.dish explant(petri.dish));
lsmeans genotype|culture/ilink lines;
run;
Some of the SAS output is shown in Table 5.21. The fit statistics in part (a) for
this dataset are shown below. Note that “method = laplace” was used for the
estimation process and to obtain Pearson’s fit statistic χ 2/DF. The result indicates
that there is evidence of overdispersion (Pearson′s chi - square/DF = 1.84).
Overdispersion, as discussed before, implies more variability in the data than
would be expected, potentially explaining the lack of fit in a Poisson model. Part
(b) shows the variance component estimates due to Petri_dish, which is equal
^2Petri:dish = 0:003616, and, for the explants within Petri.dish, it is
to σ
2
^explantðPetri:dishÞ = 0:01462. However, the type III test of fixed effects indicates that
σ
there is a statistically significant effect of genotype, culture medium, and the
interaction of both factors (part c).
The plot of residuals against the linear predictor in Fig. 5.5 provides further
evidence of possible overdispersion.
The least squares means on the model scale for the genotype (part (a)), the culture
medium (part (b)), and the interaction between both factors (part (c)) are listed under
the “Estimate” column of Table 5.22, whereas under the “Mean” column are the
means of these factors but in terms of the data.
5.2
The Poisson Model
151
Table 5.19 Number of buds counted in the cucumber experiment
Genotype
1
2
Culture
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
Table 5.20 Sources of variation and degrees of freedom
Petridish
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Explant
1
10
6
10
10
20
12
22
11
20
5
17
4
10
5
7
9
16
13
14
13
8
11
15
18
8
10
10
10
9
2
4
6
Sources of variation
Genotype
Culture
Genotype × culture
Petri.dish × Explant
Error
Total
2
7
16
12
12
16
8
13
12
18
20
10
8
4
5
5
5
9
7
6
0
9
12
6
12
12
10
10
14
5
6
12
1
3
5
20
13
2
14
18
24
18
15
0
20
5
0
6
10
8
10
3
9
3
10
8
9
6
6
12
17
14
1
2
2
9
4
7
5
0
8
18
20
15
19
18
18
12
12
5
2
2
4
11
2
9
7
9
9
9
0
4
12
10
14
9
3
0
3
5
12
16
12
15
17
20
10
14
20
5
14
10
4
3
10
4
9
3
15
5
15
12
16
13
6
15
14
9
15
7
4
5
6
1
13
15
2
20
17
14
18
18
0
21
15
8
1
0
7
12
12
18
2
9
10
16
14
11
10
12
14
9
0
3
8
Degrees of freedom
a-1=2-1=1
b-1=4-1=3
(a - 1)(b - 1) = 3
c(r - 1) = 4 × 6 - 1 = 23
(by difference) = 161
abcr - 1 = 191
152
5
Generalized Linear Mixed Models for Counts
Table 5.21 Conditional fit statistics, variance component estimates, and type III tests of fixed
effects under the Poisson distribution
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Petri.dish
Explant (Petri.dish)
(c) Type III tests of fixed effects
Effect
Num DF
Genotype
1
Culture
3
Genotype*culture
3
1168.16
354.01
1.84
Estimate
0.003616
0.01462
Den DF
161
161
161
Standard error
0.006014
0.008798
F-value
11.01
57.30
3.95
Pr > F
0.0011
<0.0001
0.0095
Since there is overdispersion in the data, we will fit the GLMM again using the
negative binomial distribution. That is, under the following GLMM:
Distribution : yij j Petri:dishk , explantðPetri:dishÞlðkÞ Negative Binomailðλij , ϕÞ,
Petri:dishj Nð0, σ2Petri:dish Þ,
explant ðPetri:dishÞlðkÞ N 0, σ2explantðPetri:dishÞ ,
Linear predictor : ηijkl = η þ αi þ βj þ ðαβÞij þ Petri:dishk þ explantðPetri:dishÞlðkÞ
Link function: log λijkl = ηijkl
and the scale parameter ϕ.
The following GLIMMIX program allows us to fit a GLMM with a negative
binomial response variable.
proc glimmix ;
class genotype culture petri.dish explant ;
model y = cultivar|culture/dist=NegBin link=log;
random petri.dish Explant(petri.dish);
lsmeans cultivar|culture;
run;
It should be noted that this program is very similar to the previous one, and the
only difference is that now a negative binomial distribution is used (“dist = negbin”).
Part of the results is presented in Table 5.23. As we have already mentioned, a
negative binomial distribution is another model for count variables when there is
overdispersion in the dataset. If Pearson’s chi-squared value divided over the degrees
of freedom is less than or equal to 1, then the overdispersion is 0 or close to 0, which
means that the model is able to efficiently capture the degree of overdispersion.
5.2
The Poisson Model
153
Fig. 5.5 Studentized conditional residuals
Based on the conditional distribution, Pearson’s chi-squared (χ 2/DF = 0.83) fit
statistic indicates that we have no evidence of overdispersion, so we can justify
the negative binomial distribution, which is better than the Poisson distribution
implemented above. In part (b), we show that the estimated scale parameter is
ϕ = 0:1712. This value is not the same as the parameter for the quasi-Poisson
model obtained with the “random _residual_” command. Note that the variance
components were slightly affected. Additionally, in Table 5.23, we can see the type
III tests for the fixed effects of the model in part (c), where a significant effect of
genotype, culture, and the interaction between both factors (genotype*culture) can
be observed on the number of buds in the leaf explant.
The “lines” option in the “lsmeans” command is used to obtain Fisher’s least
significant difference (LSD) means for both factors and their interaction. The means
and their respective standard errors, on the model scale (“Estimate” column) and on
the data scale (“Mean” column), are tabulated in Table 5.24, the genotype and
culture medium are in Table 5.25, and the interaction between both factors is in
Table 5.26. The estimated values in this mean comparison for cultivar (Table 5.24)
correspond to the values of the linear predictor ηi on the model scale, whereas the
means on the data scale is λi (part (a)) and the comparison of means (on the model
scale) are tabulated in part (b).
154
5
Generalized Linear Mixed Models for Counts
Table 5.22 Estimates on the model scale and means on the data scale under the Poisson
distribution
(a) Genotype least squares means
Standard
Genotype Estimate error
DF
1
2.2979
0.05165
161
2
2.1345
0.05298
161
(b) Culture least squares means
Standard
Culture Estimate error
DF
1
2.1984
0.06180
161
2
2.5684
0.05607
161
3
2.4609
0.05738
161
4
1.6371
0.07445
161
(c) Genotype*culture least squares means
Standard
Genotype Culture Estimate error
1
1
2.2465
0.07676
1
2
2.7789
0.06395
1
3
2.5331
0.06932
1
4
1.6331
0.09793
2
1
2.1503
0.07958
2
2
2.3580
0.07370
2
3
2.3887
0.07290
2
4
1.6411
0.09760
tvalue
44.49
40.29
tvalue
35.57
45.81
42.89
21.99
DF
161
161
161
161
161
161
161
161
Pr > |t|
<0.0001
<0.0001
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
tvalue
29.26
43.45
36.54
16.68
27.02
31.99
32.77
16.81
Mean
9.9533
8.4531
Standard error
mean
0.5141
0.4479
Mean
9.0107
13.0456
11.7156
5.1402
Standard error
mean
0.5569
0.7314
0.6723
0.3827
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
9.4547
16.1018
12.5925
5.1196
8.5877
10.5694
10.8997
5.1609
Standard
error mean
0.7258
1.0298
0.8729
0.5014
0.6834
0.7790
0.7945
0.5037
Table 5.23 Conditional fit statistics, variance component estimates, and type III tests of fixed
effects under the negative binomial distribution
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Petri.dish
Explant (Petri.dish)
Scale
(c) Type III tests of fixed effects
Effect
Num DF
Genotype
1
Culture
3
Genotype*culture
3
1143.90
159.95
0.83
Estimate
-0.02717
-0.04323
0.1712
Den DF
161
161
161
Standard error
.
.
0.03514
F-value
4.43
25.91
1.44
Pr > F
0.0369
<0.0001
0.0322
5.2
The Poisson Model
155
Table 5.24 Estimates on the model scale and means on the data scale under the negative binomial
distribution
(a) Cultivar least squares means
Standard
Genotype Estimate error
1
2.3054
0.05407
2
2.1426
0.05535
ηi
DF
161
161
tvalue
42.64
38.71
Pr > |t|
<0.0001
<0.0001
Mean
10.0287
8.5219
Standard error
mean
0.5423
0.4717
λi
(b) T grouping of genotype least squares means (α=0.05)
LS means with the same letter are not significantly different
Genotype
1
2.3054
2
2.1426
Estimate
A
B
Table 5.25 Means estimates on the model scale and data scale for the culture medium
(a) Culture least squares means
Standard
Culture Estimate error
1
2.2061
0.07653
2
2.5766
0.07198
3
2.4684
0.07300
4
1.6451
0.08708
ηj
DF
161
161
161
161
tvalue
28.82
35.80
33.81
18.89
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
(b) T grouping of culture least squares means (α=0.05)
LS means with the same letter are not significantly different
Culture
2
2.5766
3
2.4684
1
2.2061
4
1.6451
Mean
9.0802
13.1527
11.8031
5.1815
Standard error
mean
0.6950
0.9468
0.8617
0.4512
λj
Estimate
A
A
B
C
For the culture medium (Table 5.25), the estimated values in this comparison of
means correspond to the values of the linear predictor ηj (on the model scale), but, by
applying the inverse link to ηj , we obtain the values under the “Mean” column that
provide the means on the data scale (part (a)). The mean comparisons on the model
scale are shown in part (b).
The results indicate that the means in culture media 2 and 3 provided a statistically similar average number of buds compared to the means in culture media 1 and
4 (see Fig. 5.6).
The interaction between both factors (Table 5.26), the average number of buds,
and the mean comparisons are shown in Table 5.26.
156
5
Generalized Linear Mixed Models for Counts
Table 5.26 Estimates on the model scale and means on the data scale for the interaction between
genotype and culture medium
Genotype*culture least squares means
Standard
Genotype Culture Estimate error
1
1
2.2540
0.1072
1
2
2.7869
0.09844
1
3
2.5401
0.1020
1
4
1.6408
0.1233
2
1
2.1582
0.1093
2
2
2.3663
0.1050
2
3
2.3967
0.1045
2
4
1.6493
0.1230
ηij
DF
161
161
161
161
161
161
161
161
tvalue
21.02
28.31
24.91
13.31
19.75
22.53
22.94
13.41
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
9.5255
16.2310
12.6805
5.1595
8.6558
10.6582
10.9865
5.2036
Standard
error mean
1.0212
1.5978
1.2933
0.6360
0.9457
1.1196
1.1478
0.6401
λij
16
a
Average number of buds
14
a
12
10
b
8
c
6
4
2
0
1
2
3
4
Culture medium
Fig. 5.6 Comparison of the average number of buds as a function of the type of culture medium
(LSD, α = 0.05)
The values under “Estimates” (Table 5.26) correspond to those of the linear
predictor ηij (model scale), but the values under “Mean” correspond to the means
λij on the data scale.
Graphically, Fig. 5.7 shows that genotype 1 in culture medium 2 provides the
highest number of buds, whereas the lowest number of buds was observed in culture
medium 4. For genotype 2, the highest number of buds was observed in culture
media 2 and 3. Finally, culture medium 4 is less suitable for both genotypes.
5.2
The Poisson Model
157
GENOTIPO1
Average number of buds
20
a
18
16
ab
14
12
GENOTIPO2
bc
bc
bc
c
10
8
d
d
6
4
2
0
1
2
3
4
Culture medium
Fig. 5.7 Effect of the cultivar × culture medium interaction on the average number of buds
(LSD, α = 0.05)
5.2.6
Latin Square (LS) Design
A Latin square (LS) is used where heterogeneity is associated with the crossing of
two factors, generally, both with the same number of levels. This design was
originally used in agricultural experimentation with plots placed in a square arrangement, with expected heterogeneity along the rows and columns of the square.
Blocking in both directions across rows and columns is done in this experimental
design. Sometimes in experimentation, blocking in two directions may be appropriate, i.e., the use of an LS design is a good option. Some examples are provided below
to illustrate the use of this experimental design:
• Field experiments on plots set in a square arrangement with rows and columns
that contribute to the heterogeneity between plots. For example, gradients of
fertility, moisture, management practices, and so on.
• Experiments in greenhouses, rooms with a controlled environment, or growth
chambers where the placement of shelves, trays, etc. with respect to walls or light
sources can introduce systematic variability related to temperature, humidity, or
light in different directions (e.g., left to right, back to front, or top to bottom).
• Laboratory experiments in which there are two potential sources of variability
(e.g., technicians, machines, etc.) and researchers are aware of the possible impact
of variation from both sources.
For an LS layout, the number of rows (r) and columns (c) should be equal to the
number of treatments (t) and the number of replicates of each treatment. The
assignment of treatments is such that each treatment appears exactly once in each
158
5
Table 5.27 Sources of variation and degrees of freedom of
a Latin square design
Generalized Linear Mixed Models for Counts
Sources of variation
Rows
Columns
Treatments
Error
Total
Degrees of freedom
t-1
t-1
t-1
(t - 1)(t - 2)
t×t-1
row and column, with each row and column containing a full set of treatments. Thus,
the treatment effect estimates are independent of the differences between rows or
columns, and the rows, columns, and treatments are orthogonal to each other.
The analysis of variance for this experimental design, assuming that there are
r rows, c columns, and t treatments, with r = c = t, contains the following sources of
variability (Table 5.27).
From the analysis of variance table, the linear model for an LS design with
t treatments is as follows:
yijk = μ þ f j þ ck þ τi þ εijk
where yijk is the response observed in treatment i in row f and column c, μ is the
overall mean, fj is the random effect of row j assuming f j N 0, σ 2f , ck is the
random effect of column k with ck N 0, σ 2c , τi is the fixed effect of treatment i,
and εijk is the distributed random error term N(0, σ 2). Note that the treatments are
allocated in the jkth quadrant (in row j and column k).
5.2.6.1
Latin Square Design with a Poisson Response
In a series of field experiments, several “inducer-attractant” strategies were tested to
control insect pests in oilseed rape. In one experiment, the use of wild turnip rape
(turnip rape) as an earlier flowering trap crop (TR) (the “attractor”) was tested
together with the use of a repellent (an antifeedant) applied to oilseed rape in spring
(S, the “inducer”). Untreated oilseed rape (U) was included as a control. The
experiment was set up as a 6 × 6 Latin square with two replicates of each of the
three treatments per row and column. An assessment of the number of mature pollen
beetles was made on 10 plants per plot in early April, 1 day after spraying the
repellent (antifeedant). The average number of adult beetles sampled on 10 plants
per plot was recorded (Appendix 1: Data: Beatles). The question is: Is there evidence
that the attractor or inducer works? That is, are fewer beetles present in the proposed
treatments compared to the control?
The model components that define this GLMM are as described below:
5.2
The Poisson Model
159
Distribution: yijkl j f j , ck Poisson λijkl
f j N 0, σ 2f , ck N 0, σ 2c
Linear predictor: ηijkl = η þ f j þ ck þ τi
Link function: log λijkl = ηijkl
where ηijkl is the linear predictor that relates the effect of the repetition l (l = 1, 2) in
row j ( j = 1, 2, ⋯, 6) and column k (k = 1, 2, , ⋯, 6) when treatment i is applied
(i = 1, 2, 3, ), η is the intercept, τi is the fixed effect of treatment i, fj is random effect
of row j, and ck is the random effect due to column k, assuming that there is no
interaction between the rows and columns as well as between the treatments and
rows or the treatments and columns. The assumed distributions for rows and
columns are f N 0, σ 2f and ck N 0, σ 2c , respectively. The model uses the
linear predictor (ηijkl) to estimate the means (λijkl = μijkl) of the treatments.
The following GLIMMIX program fits a Latin square design with a Poisson
response:
Proc glimmix nobound method=laplace;
class Row Column Treatment;
model count = treatment/dist=Poi link=log;
random row column;
lsmeans treatment/lines ilink;
run;
Part of the output is shown in Table 5.28. In the values of the fit statistics (part
(a)), we observe that the value of Pearson’s chi-square divided by the degrees of
χ2
= 0:55 , indicating that there is no overdispersion in the
freedom is less than 1 DF
data and that the Poisson distribution adequately models the dataset.
The type III tests of fixed effects in part (b) indicate that there is no significant
evidence of differences between the treatments (P = 0.0621).
Part (c) of Table 5.28 shows the estimates of treatments on the model scale
(“Estimate”) and on the data scale (“Mean”) with their respective standard errors.
The values 4.6191, 6.9396, and 5.1561 (under the “Mean” column) correspond to
the treatment means for S, TR, and U, respectively.
5.2.6.2
Randomized Complete Block Design in a Split Plot
Sometimes the researcher is interested in testing multiple factors using different
experimental units, and, in most cases, the experimenter cannot randomly accommodate the treatment combinations. Suppose that one wishes to test two factors, A
and B with a and b levels each, respectively. The levels of the first factor (A) are
randomly applied to the primary experimental units. Then, the levels of the second
160
5 Generalized Linear Mixed Models for Counts
Table 5.28 Results of the analysis of variance
(a) Fit statistics for conditional distribution
-2 Log L (Conteo | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
Treatment
3
23
(c) Treatment least squares means
Standard
tTreatment Estimate error
DF value
S
1.5302
0.1343
23 11.39
TR
1.9372
0.1096
23 17.68
U
1.6402
0.1271
23 12.90
τi
147.03
19.78
0.55
F-value
3.14
Pr > |t|
<0.0001
<0.0001
<0.0001
Mean
4.6191
6.9396
5.1561
Pr > F
0.0621
Standard error
mean
0.6204
0.7605
0.6555
λi
factor (B) are applied to the secondary subunits formed within the primary unit in
which the first factor was applied. In other words, the primary experimental unit
(whole plot) was used for the application of the first factor; then, after this, it was
divided to form the secondary experimental units (subplots) for the application of the
levels of the second factor. Since the split-plot design has two levels of experimental
units, the whole plot portions (primary units) and subplots (secondary units) have
different experimental errors. Split-plot experiments were invented in agriculture by
Fisher (1925), and their importance in industrial experimentation has been widely
recognized (Yates 1935).
As a simple illustration, consider a study of three pulp preparation methods
(factor A) and four temperature levels (factor B) on the effect of paper tensile
strength (paper quality). A batch of pulp is produced by one of the three methods;
it is then divided into four equal portions (samples). Each portion is cooked at a
specific level of temperature. The assignment of treatments to plots and subplots is
shown in Table 5.29.
The standard ANOVA model for two factors in a split-plot design, in which there
are three levels of factor A and four levels of factor B nested within factor A, is
described below:
yijk = μ þ αi þ r k þ αðr Þik þ βj þ ðαβÞij þ εijk
where yijk is the observed response at level i (i = 1, 2, 3) of factor A and at level
j ( j = 1, 2, 3, 4) of factor B in block k (k = 1, 3, 3), μ is the overall mean, αi is the
effect at level i of factor A, rk is the random effect of blocks assuming rk N 0, σ 2r ,
α(r)ik is the random effect of the error of the whole plot assuming
αðr Þik N 0, σ 2αðrÞ , βj is the effect at level j of factor B, (αβ)ij is the interaction
5.2
The Poisson Model
161
Table 5.29 Assigning treatments to whole plots and subplots
Block 1
Preparation method
A1
A3
A2
AB14
AB34
AB24
AB12
AB32
AB22
AB11
AB31
AB21
AB13
AB33
AB23
Temperature
B4
B2
B1
B3
Table 5.30 Sources of variation and degrees of freedom
for a randomized block design
with a split-plot treatment
arrangement
Block 2
Preparation method
A3
A2
A1
AB34
AB24
AB14
AB32
AB22
AB12
AB31
AB21
AB11
AB33
AB23
AB13
Sources of variation
Blocks
Factor A
Errora
Factor B
A×B
Error
Total
Block 3
Preparation method
A2
A3
A1
AB24
AB34
AB14
AB22
AB32
AB12
AB21
AB31
AB11
AB23
AB33
AB13
Degrees of freedom
r-1=3-1=2
a-1=3-1=2
(a - 1)(r - 1) = 4
b-1=4-1=3
(a - 1)(b - 1) = 6
a(r - 1)(b - 1) = 3 × 2 × 3 = 18
r × a × b - 1 = 3 × 3 × 4 - 1 = 35
fixed effect at level i of factor A and at level j of factor B, and εijk is the normal
random experimental error {εijk~iidN(, σ 2)}. The ANOVA table with sources of
variation is shown in Table 5.30 for this experimental design.
Example 5.1 A split-plot design in randomized complete block arrangement with a
Poisson response
A split plot is probably the most common design structure in plant and soil
research. Such experiments involve two or more treatment factors. Typically, large
units called whole plots are grouped into blocks. The levels of the first factor are
randomly assigned to whole plots. Each whole plot is divided into smaller units,
called subplots (split plots). Next, the levels of the second factor are randomly
assigned to units of split plots within each whole plot.
In this example, four blocks were implemented, which were divided into seven
parts for the seven levels of the first factor (A1, A2, A3, A4, A5, A6, and A7), as whole
plots. Then, each whole plot was divided into four units for randomly assigning the
four levels of factor B, known as subplots (B1, B2, B3, and B4). Both factors were
used to control the growth of a particular weed. Both factors were randomly
allocated in each block, as shown below:
Block 1
A7
A1
B3
B3
B1
B2
B2
B4
B4
B1
A3
B4
B3
B1
B2
A2
B1
B3
B4
B2
A5
B2
B1
B3
B4
A4
B1
B2
B3
B4
A6
B3
B2
B4
B1
⋯
⋯
⋯
Block 4
A6
A3
B3
B3
B1
B2
B2
B4
B4
B1
A7
B4
B3
B1
B2
A2
B1
B3
B4
B2
A1
B2
B1
B3
B4
A5
B1
B2
B3
B4
A4
B3
B2
B4
B1
162
5
Table 5.31 Sources of variation and degrees of freedom
for a randomized block design
with a split-plot treatment
arrangement
Generalized Linear Mixed Models for Counts
Sources of variation
Blocks
Factor A
Errora(A × r)
Factor B
A×B
Errorb
Total
Degrees of freedom
r-1=4-1=3
a-1=7-1=6
(a - 1)(r - 1) = 18
b-1=4-1=3
(a - 1)(b - 1) = 18
a(r - 1)(b - 1) = 7 × 3 × 3 = 63
r × a × b - 1 = 4 × 7 × 4 - 1 = 111
The sources of variation and degrees of freedom for this experiment are shown
below in Table 5.31:
In this experiment, the response variable was the number of weeds in each of the
plots (Appendix 1: Weed counts). The components that define this GLMM are as
shown below:
Distribution: yijk j r k , αðr Þik Poisson λijk
r k N 0, σ 2r , αðr Þik N 0, σ 2ar
Linear predictor: ηijk = η þ αi þ r k þ αðr Þik þ βj þ ðαβÞij
Link function: log λijk = ηijk
where ηijk is the linear predictor that relates the effect of factor A with i levels
(i = 1, 2, ⋯, 7)and factor B with j levels ( j = 1, 2, 3, 4) in block k with
(k = 1, 2, 3, 4); η is the intercept, αi is the fixed effect at level i of factor A, βj is
the fixed effect at level j of factor B, (αβ)ij is the fixed effect of the interaction
between level i of factor A and level j of factor B, rk is the random effect due to
block; and α(r)ik is the random error effect of the whole plot, assuming r k N 0, σ 2r and αðr Þik N 0, σ 2AR , respectively. The model uses the aforementioned
linear predictor (ηijk) to estimate the means (λijk = μijk) of the treatments.
The following GLIMMIX program fits a split-plot block design with a Poisson
response variable:
proc glimmix method=laplace;
class block a b;
model count=a|b / dist=Poisson link=log;
random block block*a;
lsmeans a|b /lines ilink;
run;
Part of the output is shown below.
As in the previous examples, the Poisson model was found to be inadequate
because the value of Pearson’s chi-squared statistic divided by the degrees of
5.2
The Poisson Model
163
Table 5.32 Results of the
analysis of variance
(a) Fit statistics for conditional distribution
-2 Log L (Conteo | r. effects)
1053.96
Pearson’s chi-square
504.44
Pearson’s chi-square/DF
4.50
(b) Covariance parameter estimates
Cov Parm
Estimate
Standard error
Bloque
0.01526
0.03867
Bloque*A
0.2454
0.07565
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
A
6
18
2.32
0.0775
B
3
63
22.91
<0.0001
A*B
18
63
10.06
<0.0001
freedom is greater than 1
χ2
df
= 4:50 .This indicates that we have probably
misspecified either the conditional distribution of y j b or the linear predictor, but,
in this case, there is evidence that we need to look for other distributions for this
dataset (part (a), Table 5.32. In addition, in part (b), the values of variance component estimates due to blocks and blocks × A are tabulated
σ^2r = 0:01526; σ^2ra = 0:2454 . On the other hand, the type III tests of fixed effects
(part (c)) show a significant effect of factor B and the interaction between both
factors.
An alternative to reduce the overdispersion is to keep the same linear predictor,
changing the Poison distribution in the response variable by the negative binomial
distribution, that is:
Distribution: yijk j r k , αðr Þik Negative binonial λijk , ϕ
r k iidN 0, σ 2r , αðr Þik iidN 0, σ 2AR
Linear predictor: ηijk = η þ αi þ r k þ αðr Þik þ βj þ ðαβÞij
Link function: log λijk = ηijk
The following syntax fits a GLMM under a negative binomial distribution.
proc glimmix method=Laplace;
class block a b;
model count=a|b / dist=NegBin link=log;
random intercept a /subject=block;
lsmeans a|b/lines ilink;
run;
Part of the output is shown below (Table 5.33). According to the results tabulated
in (a), they indicate that the overdispersion has been removed from the analysis
164
5
Table 5.33 Results of the
analysis of variance
χ2
df
Generalized Linear Mixed Models for Counts
(a) Fit statistics for conditional distribution
-2 log L (Conteo | r. effects)
838.51
Pearson’s chi-square
79.36
Pearson’s chi-square/DF
0.71
(b) Covariance parameter estimates
Cov Parm
Subject
Estimate
Standard error
Intercept
Bloque
0.002421
0.02768
A
Bloque
0.1222
0.07102
Scale
0.3458
0.06875
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
A
6
18
2.71
0.0473
B
3
63
2.13
0.1054
A*B
18
63
1.18
0.3017
= 0:71 . The variance components estimates, tabulated in part (b), are σ 2r =
0:0024 and σ 2AR = 0:1222 for blocks and blocks × A, respectively. The estimated
scale parameter is ϕ = 0:3458. Note that the results under the negative binomial
distribution differ from those obtained under the Poisson distribution, which is due,
of course, to the fact that the negative binomial distribution better captures
overdispersion. The fixed effects F-test for factor A is significant at the 5% significance level (part (c)), whereas factor B and the interaction effect do not significantly
influence the response variable.
Example 5.2 A split-split plot in time in a randomized complete block design with a
Poisson response.
The propagation of coffee seedlings through grafting in nurseries depends on
several factors such as the type of substrate, the rootstock of the plant that will host
the graft, type of graft, light intensity, type and size of the container, humidity,
temperature, and so forth. The objective of this experiment was to evaluate the effect
of shade cloth (light intensity), type of container, and clone on the number of leaves
produced by the Coffea canephora P. clones grafted with the Coffea arabica
L. variety Oro azteca.
The factors studied were the color of the shade cloth (black, pearl, and red),
container size (tube of 0.5 kg and 1 kg), and five coffee clones of the variety Coffea
canephora P. plus a franc foot (Coffea arabica L. and Var. Oro azteca) over a period
of 11 months (Appendix 1: Coffee data). The clones used in the experiment are listed
below (Table 5.34). Different physiological parameters were evaluated for
11 months.
This work was implemented in four randomized complete blocks. The following
table exemplifies how a block was constructed.
5.2
The Poisson Model
165
Table 5.34 Clones of Coffea canephora P
Graft carrier (Coffea canephora P.)
Clone 1
Clone 2
Clone 3
Clone 4
Clone 5
Franc foot: Coffea arabica L. Var. Aztec gold
Shade cloth red
Container
Tray 0.5 kg
C2
C5
C4
Pf
C3
C1
C5
C2
Pf
C4
C1
C3
Container
1 kg
C4
C3
C5
Pf
C1
C2
Grafting (Coffea arabica L.)
Var. Aztec gold
Var. Aztec gold
Var. Aztec gold
Var. Aztec gold
Var. Aztec gold
Shade cloth Perl
Container
Tray 0.5 kg
C4
C5
C3
Pf
C5
C1
Pf
C2
C1
C4
C2
C3
Container
1 kg
C2
C4
C3
C5
Pf
C1
Shade cloth black
Container
Tray 0.5 kg
C5
C2
Pf
C3
C1
C5
C2
Pf
C4
C1
C3
C24
Code
C1
C2
C3
C4
C5
Pf
Container
1 kg
C4
C2
C3
C5
Pf
C1
The statistical model describing a split-split plot in time design is described
below:
yijklm = μ þ αi þ r m þ ðar Þim þ βj þ ðαβÞij þ γ k þ ðαγ Þik þ ðβγ Þjk þ ðαβγ Þijk
þðrabγ Þijkm þ τl þ ðατÞil þ ðβτÞjl þ ðαβτÞij þ ðγτÞkl þ ðαγτÞikl
þðβγτÞjkl þ ðαβγτÞijkl þ εijklm
i = 1, 2, 3; j = 1, 2, 3, 4, 5; k = 1, 2, 3; l = 1, ⋯, 11; m = 1, 2, 3, 4
where yijklm is the response variable in repetition m, shade cloth i, clone j, and tray
k in time l; μ is the overall mean; αi is the fixed effect due to the type of shade cloth;
βj, γ k, and τl are the fixed effects due to clone type, tray,and sampling time,
respectively; (αβ)ij, (αγ)ik,(βγ)jk, (ατ)il, (βτ)jl, and (γτ)kl are the effects of the double
interactions of the factors shade cloth type with clone, tray, and sampling time;
(αβγ)ijk, (αβτ)ij, (αγτ)ikl, (βγτ)jkl, and (αβγτ)ijkl are the effects of the third and fourth
interactions of the factors under study; (ar)im is the random effect of blocks with type
of shade cloth with rm, (ar)im, (rabγ)ijkm are the random effect due to blocks, blocks
with type of shade cloth, blocks with type of shade cloth, and time assuming
r m N 0, σ 2r , ðar Þim N 0, σ 2rα ðrabγ Þijkm N 0, σ 2αβγðrepÞ , and εijklm is random
error {εijklm~N(0, σ 2)}.
The following SAS program fits a GLMM in a split-split plot in time under a
randomized complete block design with a Poisson response.
proc glimmix data=work.Nhojas_cafe nobound method=laplace;
class shade clone tray rep time;
166
5
Generalized Linear Mixed Models for Counts
Table 5.35 Fit statistics for choosing the correlation structure
Correlation structure
Fit statistics
-2 Log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
CS
29047.38
30236.38
30338.31
29871.61
30469.61
29435.72
Table 5.36 Conditional fit
statistics and variance component estimates
AR(1)
29043.38
30235.38
30337.31
29869.61
30465.61
29432.72
UN
Not converged
TOEP(1)
29053.85
30243.85
30345.44
29878.70
30473.70
29442.55
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Subject
Estimate
Variance
Rep
0.008106
AR(1)
Rep
-0.3254
ANTE(1)
No converged
28709.17
4288.74
0.56
Standard error
0.001392
0.09437
model y = shade|clone|tray|time/dist=poi link=log;
random intercept shade shade*clone*tray/subject=rep type=ar(1) ;
lsmeans shade|clone|tray|time /lines ilink;
run;
Some of the results are listed below. To study which correlation structure best fits
this experimental design, five types of correlation structures were tested
(Table 5.35): compound symmetry (“CS”), autoregression of order 1 (“AR(1)”),
unstructured (“UN”), Toeplizt of order 1 (“Toep(1)”), and ante (ANTE(1)). To do
this, in the “random” command with the “type” option, the type of correlation to be
tested is specified, and it is here where the option of type of variance–covariance
structure must be changed. The fit statistics indicate that the variance–covariance
structure that best fits the model is the autoregressive structure of order 1 hAR(1)i.
This can be seen in the following table in which the goodness-of-fit statistics for
choosing between all these variance–covariance structures are reported.
Table 5.36 shows the conditional statistics and variance component estimates.
The fit statistic Pearson′s chi - square/DF = 0.57 in part (a) indicates that, in a
conditional model, there is no evidence of mis-specifying the distribution or linear
predictor. In other words, there is no overdispersion in the dataset, and, therefore, it
is reasonable that the analysis and inference can be based on the Poisson model.
The analysis of variance for the type III tests of fixed effects (Table 5.37)
indicates that there is a highly significant effect of the main effect type of shade
cloth (P = 0.0001), clone (P = 0.0001), and tray (P = 0.0001) as well as of most of
the interactions, except for the interactions shade_cloth*clone; (P = 0.3846),
5.2
The Poisson Model
167
Table 5.37 Type III fixed effects tests
Type III tests of fixed effects
Effect
Shade
Clone
Shade*clone
Tray
Shade*tray
Clone*tray
Shade*clone*tray
Time
Shade*time
Clone*time
Shade*clone*time
Tray*time
Shade*tray*time
Clone*tray*time
Shade*clone*tray*time
Num DF
2
5
10
2
4
10
20
10
20
50
100
20
40
100
200
Den DF
6
153
153
153
153
153
153
6822
6822
6822
6822
6822
6822
6822
6822
Pr > F
0.1011
<0.0001
0.3846
<0.0001
<0.0001
0.0027
0.0363
<0.0001
<0.0001
<0.0001
0.9289
<0.0001
<0.0001
0.9760
0.2484
F-value
3.44
16.38
1.08
56.60
8.83
2.86
1.71
721.20
6.91
3.17
0.80
9.03
2.42
0.74
1.07
Table 5.38 Estimated means on the model scale and on the data scale for the shade cloth
(a) Shade cloth least squares means
Shade
Standard
Estimate error
DF t-value Pr > |t|
cloth
Black
1.6221
0.01542
2
105.17 <0.0001
Pearl
1.5472
0.01533
2
100.94 <0.0001
Red
1.7184
0.01301
2
132.09 <0.0001
(b) T grouping of shade cloth least squares means (α=0.05)
LS means with the same letter are not significantly different
Shade cloth
Estimate ðτi Þ
Red
1.7184
Black
1.6221
1.5472
Pearl
Mean
5.0638
4.6981
5.5757
Standard error
mean
0.07810
0.07201
0.07254
A
B
B
shade_cloth*tray*time (P = 0.9289), clone*tray*time (P = 0.9760), and
shade_cloth*clone*tray*time (P = 0.2484).
The means and standard errors of each of the main effects, on the data scale, for
shade_cloth, tray, and clone are shown in the “Mean” column in part (a) of
Table 5.38, whereas in part (b), the mean comparisons for the type of shade cloth
are shown.
Table 5.39 presents the estimates of the linear predictor (“Estimates” column) in
terms of the model scale and treatment means in terms of the data scale (“Mean”
column) for the type of clone (part (a)). In addition, in Table 5.39 (part (b)), the mean
comparisons are presented for the type of clone.
168
5
Generalized Linear Mixed Models for Counts
Table 5.39 Estimated means on the model scale and on the data scale for the type of clone
(a) Clone least squares means
Clone Estimate Standard error DF t-value Pr > |t|
C1
1.5008
0.04989
153 30.08
<0.0001
C2
1.4250
0.05080
153 28.05
<0.0001
C3
1.5064
0.05019
153 30.02
<0.0001
C4
1.4750
0.05029
153 29.33
<0.0001
C5
1.5965
0.04970
153 32.12
<0.0001
Pf
1.6344
0.04943
153 33.07
<0.0001
(b) T grouping of clone least squares means (α=0.05)
LS means with the same letter are not significantly different
Clone
Estimate
Pf
1.6344
C5
1.5965
C3
1.5064
C1
1.5008
C4
1.4750
C
C2
1.4250
C
Mean
4.4854
4.1578
4.5106
4.3709
4.9357
5.1264
Standard error mean
0.2238
0.2112
0.2264
0.2198
0.2453
0.2534
A
A
B
B
B
Table 5.40 Estimated means on the model scale and on the data scale for the tray factor
(a) Tray least squares means
Tray Estimate Standard error DF t-value Pr > |t|
CH1 1.3843
0.04859
28.49
<0.0001
CH2 1.5665
0.04838
32.38
<0.0001
CH3 1.6183
0.04819
33.58
<0.0001
(b) T grouping of tray least squares means (α=0.05)
LS means with the same letter are not significantly different
Tray
CH3
1.6183
CH2
1.5665
CH1
1.3843
Mean
3.9921
4.7898
5.0443
Standard error mean
0.1940
0.2317
0.2431
Estimate
A
B
C
Table 5.40 presents the estimates for the levels of the tray on both scales (part (a)).
Similarly, in this table (part (b)), the treatment mean comparisons are presented for
the levels of the tray.
Tables 5.41, 5.42, 5.43, and 5.44 show the means and standard errors on both
scales of the two-factor and three-factor interactions.
Interaction type of shade cloth*clone
Interaction type of shade cloth*tray
Interaction clone*tray
Interaction shade*clone*tray
Although it is not the objective of this book, part of the results is discussed below.
In Fig. 5.8, it is possible to observe that the red shade cloth significantly stimulates
leaf production in coffee grafts, followed by the black and pearl shade cloths. The
5.2
The Poisson Model
169
Table 5.41 Estimated means on the model scale and on the data scale for the type of shade
cloth*clone
Shade cloth*clone least squares means
Shade
Standard
cloth
Clone Estimate error
Black
C1
1.5109
0.06230
Black
C2
1.3340
0.06507
Black
C3
1.4990
0.06354
Black
C4
1.4485
0.06425
Black
C5
1.5916
0.06163
Black
pf
1.6219
0.06114
Pearl
C1
1.3835
0.07711
Pearl
C2
1.3926
0.07781
Pearl
C3
1.4028
0.07589
Pearl
C4
1.4288
0.07575
Pearl
C5
1.5216
0.07536
Pearl
pf
1.5285
0.07458
Red
C1
1.6081
0.06991
Red
C2
1.5483
0.07100
Red
C3
1.6175
0.07055
Red
C4
1.5477
0.07072
Red
C5
1.6762
0.06971
Red
pf
1.7528
0.06923
DF
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
tvalue
24.25
20.50
23.59
22.54
25.83
26.53
17.94
17.90
18.49
18.86
20.19
20.50
23.00
21.81
22.93
21.88
24.04
25.32
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
4.5307
3.7961
4.4771
4.2566
4.9118
5.0628
3.9889
4.0254
4.0666
4.1736
4.5797
4.6112
4.9933
4.7036
5.0404
4.7005
5.3451
5.7707
Standard error
mean
0.2823
0.2470
0.2845
0.2735
0.3027
0.3095
0.3076
0.3132
0.3086
0.3161
0.3451
0.3439
0.3491
0.3339
0.3556
0.3324
0.3726
0.3995
Table 5.42 Estimated means on the model scale and on the data scale for the interaction type of
shade cloth*tray
Shade*tray least squares means
Standard
Shade
Tray Estimate error
cloth
Black
CH1 1.4274
0.05961
Black
CH2 1.5523
0.05846
Black
CH3 1.5232
0.05824
Pearl
CH1 1.2070
0.07354
Pearl
CH2 1.4972
0.07218
Pearl
CH3 1.6247
0.07145
Red
CH1 1.5185
0.06733
Red
CH2 1.6499
0.06714
Red
CH3 1.7068
0.06732
DF
153
153
153
153
153
153
153
153
153
tvalue
23.94
26.55
26.15
16.41
20.74
22.74
22.55
24.57
25.35
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
4.1679
4.7224
4.5869
3.3434
4.4691
5.0771
4.5655
5.2066
5.5114
Standard error
mean
0.2485
0.2761
0.2672
0.2459
0.3226
0.3628
0.3074
0.3496
0.3710
production of leaves in coffee grafts shows a bimodal figure that can be due to factors
such as humidity and temperature. Extreme conditions of both factors cause stress at
the growing points and, therefore, the appearance of leaves.
Regarding the type of clone used as rootstock, the clones showed a better average
leaf production in months 5 and 6, whereas the lowest production was observed in
170
5
Generalized Linear Mixed Models for Counts
Table 5.43 Estimated means on the model scale and on the data scale for the clone–tray interaction
Clone*tray least squares means
Standard
Clone Tray Estimate error
C1
CH1 1.3916
0.06112
C1
CH2 1.5502
0.05861
C1
CH3 1.5607
0.05861
C2
CH1 1.2242
0.06459
C2
CH2 1.4780
0.06029
C2
CH3 1.5727
0.05890
C3
CH1 1.2924
0.06114
C3
CH2 1.5433
0.05975
C3
CH3 1.6836
0.05841
C4
CH1 1.2982
0.06251
C4
CH2 1.5829
0.05815
C4
CH3 1.5439
0.05939
C5
CH1 1.5311
0.05843
C5
CH2 1.5981
0.05920
C5
CH3 1.6602
0.05803
pf
CH1 1.5684
0.05794
pf
CH2 1.6464
0.05833
pf
CH3 1.6884
0.05728
DF
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
tvalue
22.77
26.45
26.63
18.95
24.51
26.70
21.14
25.83
28.83
20.77
27.22
26.00
26.20
26.99
28.61
27.07
28.23
29.48
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
4.0214
4.7122
4.7622
3.4014
4.3843
4.8196
3.6414
4.6799
5.3851
3.6626
4.8690
4.6828
4.6234
4.9438
5.2604
4.7989
5.1884
5.4107
Standard error
mean
0.2458
0.2762
0.2791
0.2197
0.2644
0.2839
0.2226
0.2796
0.3145
0.2289
0.2831
0.2781
0.2702
0.2927
0.3053
0.2781
0.3026
0.3099
months 1, 2, 8, and 9. The franc foot showed a higher average of leaves compared to
the rest of the clones (Fig. 5.9).
5.3
Exercises
Exercise 5.3.1 A researcher in the area of plant sciences wants to know what is the
response of a plant in vitro culture when it is exposed to different concentrations
(ppm) of a chemical compound to the number of outbreaks that the explant produces
(yij). The data for this experiment are given below (Table 5.45):
(a) Write down the analysis of variance table (sources of variation and degrees of
freedom).
(b) Write down the components of the GLMM.
(c) Analyze the dataset with the model proposed in (b).
(d) Compare and contrast the results of these analyses. If necessary, reanalyze the
dataset using the same model as above, but, now, assume that the data have a
negative binomial distribution.
(e) Summarize the relevant results.
5.3
Exercises
171
Table 5.44 Estimated means on the model scale and on the data scale for the shade–clone–tray
interaction
Shade*clone*tray least squares means
Shade
Standard
cloth
Clone Tray Estimate error
Black C1
CH1 1.2821
0.1528
Black C1
CH2 1.4143
0.1521
Black C1
CH3 1.2201
0.1538
Black C2
CH1 0.8131
0.1615
Black C2
CH2 1.1486
0.1543
Black C2
CH3 1.3376
0.1533
Black C3
CH1 1.1809
0.1548
Black C3
CH2 1.1105
0.1550
Black C3
CH3 1.3672
0.1528
Black C4
CH1 0.7672
0.1608
Black C4
CH2 1.4660
0.1517
Black C4
CH3 1.3925
0.1523
Black C5
CH1 1.2316
0.1538
0.1507
Black C5
CH2 1.6090
Black C5
CH3 1.4684
0.1515
Black Pf
CH1 1.6751
0.1503
Black Pf
CH2 1.3126
0.1548
Black Pf
CH3 1.5092
0.1511
Pearl
C1
CH1 0.6441
0.1741
Pearl
C1
CH2 1.3602
0.1639
Pearl
C1
CH3 1.6030
0.1633
Pearl
C2
CH1 0.6336
0.1741
Pearl
C2
CH2 1.2050
0.1672
Pearl
C2
CH3 1.5547
0.1635
Pearl
C3
CH1 0.8786
0.1684
Pearl
C3
CH2 1.2777
0.1646
Pearl
C3
CH3 1.5724
0.1637
Pearl
C4
CH1 0.9893
0.1680
Pearl
C4
CH2 1.4198
0.1636
Pearl
C4
CH3 1.4357
0.1646
Pearl
C5
CH1 1.4557
0.1631
Pearl
C5
CH2 1.1672
0.1696
Pearl
C5
CH3 1.6010
0.1633
Pearl
Pf
CH1 1.1901
0.1649
Pearl
Pf
CH2 1.4004
0.1643
Pearl
Pf
CH3 1.7623
0.1620
Red
C1
CH1 1.5245
0.1606
Red
C1
CH2 1.6004
0.1605
Red
C1
CH3 1.6327
0.1607
Red
C2
CH1 1.3462
0.1630
DF
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
153
tvalue
8.39
9.30
7.93
5.04
7.45
8.72
7.63
7.17
8.95
4.77
9.66
9.14
8.01
10.67
9.70
11.15
8.48
9.99
3.70
8.30
9.82
3.64
7.21
9.51
5.22
7.76
9.60
5.89
8.68
8.72
8.93
6.88
9.80
7.22
8.52
10.88
9.49
9.97
10.16
8.26
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
0.0003
<0.0001
<0.0001
0.0004
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
3.6041
4.1136
3.3874
2.2549
3.1538
3.8100
3.2574
3.0359
3.9242
2.1538
4.3318
4.0250
3.4269
4.9979
4.3422
5.3393
3.7160
4.5231
1.9043
3.8970
4.9678
1.8844
3.3366
4.7335
2.4074
3.5885
4.8184
2.6893
4.1362
4.2026
4.2875
3.2130
4.9582
3.2875
4.0570
5.8260
4.5930
4.9548
5.1178
3.8430
Standard
error mean
0.5509
0.6258
0.5209
0.3641
0.4866
0.5842
0.5041
0.4705
0.5996
0.3462
0.6573
0.6131
0.5270
0.7534
0.6577
0.8025
0.5753
0.6834
0.3314
0.6387
0.8111
0.3281
0.5579
0.7740
0.4053
0.5905
0.7889
0.4519
0.6769
0.6919
0.6992
0.5449
0.8098
0.5422
0.6665
0.9440
0.7379
0.7953
0.8224
0.6264
(continued)
172
5
Generalized Linear Mixed Models for Counts
Table 5.44 (continued)
Shade*clone*tray least squares means
Shade
Standard
Clone Tray Estimate error
cloth
Red
C2
CH2 1.4270
0.1622
Red
C2
CH3 1.6500
0.1620
Red
C3
CH1 1.3915
0.1632
Red
C3
CH2 1.4491
0.1614
Red
C3
CH3 1.7875
0.1603
Red
C4
CH1 1.3961
0.1614
Red
C4
CH2 1.5874
0.1606
Red
C4
CH3 1.3805
0.1635
Red
C5
CH1 1.6313
0.1601
Red
C5
CH2 1.6395
0.1605
Red
C5
CH3 1.6470
0.1610
Red
Pf
CH1 1.7075
0.1600
Red
Pf
CH2 1.7594
0.1594
0.1601
Red
Pf
CH3 1.7568
DF
153
153
153
153
153
153
153
153
153
153
153
153
153
153
tvalue
8.80
10.19
8.53
8.98
11.15
8.65
9.89
8.44
10.19
10.22
10.23
10.67
11.04
10.98
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
4.1663
5.2071
4.0207
4.2592
5.9747
4.0394
4.8910
3.9768
5.1103
5.1527
5.1912
5.5151
5.8087
5.7938
Standard
error mean
0.6759
0.8435
0.6563
0.6872
0.9577
0.6520
0.7854
0.6503
0.8180
0.8268
0.8360
0.8825
0.9260
0.9273
Avergage number of leafs
12.5
10.
7.5
5.
2.5
0.
1
2
3
4
5
6
7
Time (months)
Black
Pesrl
Red
Fig. 5.8 Effect of mesh type on the average number of leaves
8
9
10
11
5.3
Exercises
173
Average number of leafs
11.
8.25
5.5
2.75
0.
1
2
3
4
5
6
7
8
9
10
11
Time (months)
Clone1
Clone2
Clone3
Clone4
Clone5
Pf
Fig. 5.9 Effect of mesh type on the average number of leaves
Exercise 5.3.2 Earthworms (Lubricus terrestris L.) were counted in four replicates
of a factorial experiment at the W.K. Kellogg Biological Station in Battle Creek,
Michigan, in 1995. A 24 factorial experiment was conducted. Factors and treatment
levels were plowing (chiseled and unplowed), input level (conventional and low),
manure application (yes/no), and crop (corn and soybean). The objective of interest
was whether L. terrestris density varies according to these management protocols
and how various factors act and interact. The data (not pooled) in the table shows the
total worm counts (per square foot) in the factorial design 24 for the experimental
units 64 (24 × 4) (juvenile and adult worms). The numbers in each cell of the table
correspond to the counts in the replicates (Table 5.46).
(a) Write down the analysis of variance table (sources of variation and degrees of
freedom).
(b) Write down the components of the GLMM.
(c) Analyze the dataset with the model proposed in (b).
(d) Summarize the relevant results.
Exercise 5.3.3 This experiment involves an investigation of genotypic variation
within cultivars of pore (Allium porrum L.) with respect to adventitious shoot
formation in the callus tissue. The data in Table 5.47 refer to 20 genotypes of
1 cultivar. Each genotype is represented by six calluses. These observations are
the number of shoots per callus. The data are subject to two sources of variation, i.e.,
variation between genotypes and variation between the calluses within the
genotypes.
174
5
Generalized Linear Mixed Models for Counts
Table 5.45 In vitro culture (Conc = concentration in ppm)
Conc
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
50
50
50
50
50
50
50
50
50
50
50
50
Explant
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
5
6
7
8
9
10
11
12
No. of outbreaks
39
32
36
46
30
40
46
28
29
25
29
36
28
35
35
45
45
38
34
47
36
47
35
38
39
42
42
41
31
33
37
38
54
45
57
38
60
35
45
44
33
49
58
45
Conc
50
50
50
50
50
50
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
Explant
13
15
16
17
18
19
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
No. of outbreaks
54
35
50
51
38
61
46
55
54
49
55
55
47
42
38
50
46
42
44
30
38
31
42
36
37
27
38
25
29
30
30
37
28
37
29
36
34
27
32
37
30
31
30
5.3
Exercises
175
Table 5.46 Results of the experiment with earthworms
Cultivation
Manure
Corn
Yes
No
Yes
No
Soy
Table 5.47 Results of the
callus tissue experiment
Tillage
Chisel ploughing
Entry level
Low
Conventional
5, 5, 4, 2
5, 1, 5, 0
3, 11, 0, 0
2, 0, 6, 1
8, 6, 0, 3
8, 4, 2, 2
8, 5, 3, 11
2, 6, 9, 4
Callus
Genotype
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
0
9
2
1
6
6
0
1
3
3
2
0
9
2
0
5
1
0
1
4
2
0
0
4
2
3
2
2
1
3
6
6
0
3
3
0
4
0
1
4
3
No Tillage
Entry level
Low
8, 4, 6, 4
2, 2, 11, 4
2, 2, 13, 7
7, 5, 18, 3
3
0
1
4
5
8
4
0
3
1
4
8
3
5
2
0
4
0
0
6
5
Conventional
14, 9, 9, 6
15, 9, 6, 4
5, 3, 6, 0
23, 12, 17, 9
4
5
0
5
0
9
3
4
0
1
0
7
8
2
5
0
0
7
0
0
2
18
3
2
4
0
5
2
1
0
6
1
7
10
6
3
1
7
0
1
0
4
6
0
4
0
4
9
7
0
2
2
8
5
6
4
2
1
1
1
0
7
0
(a) Write down the analysis of variance table (sources of variation and degrees of
freedom).
(b) Write down the components of the GLMM.
(c) Analyze the dataset with the model proposed in (b).
(d) Reanalyze the dataset using the same model as above, but, now, assume that the
data have a negative binomial distribution.
(e) Compare and contrast the results of these analyses.
(f) Summarize the relevant results.
176
5
Generalized Linear Mixed Models for Counts
Exercise 5.3.4 In an experiment at the Research Institute for Animal Production
“Schoonoord” in the Netherlands, the effects of active immunization against androstenedione on the fertility of Texel ewes were studied (Engel and te Brake 1993).
The number of fetuses per ewe can be considered as the net result of a process that
determines the number of ovulations and a probability process for these ovulations to
produce fetuses. In this study, the goals are to model and analyze (a) the number of
ovulations and the number of fetuses in relation to Fecundin (androstenedione-7acarboxyethylthioether) treatment, animal age, mating period and (b) the number of
fetuses in relation to treatment, animal age, and number of ovulations observed. A
summary of the experiment and a summary of the data are shown below
(Table 5.48).
Of the 125 Texel ewes, 63 are treated with Fecundin, whereas the remaining
62 serve as a control group. The ewes are sorted into four age classes (e.g.,<0.5,
0.5 - 1.5, 1.5 - 2.5, and > 2.5 years) and two mating periods (starting on October
1 and October 22, 1986, respectively). The interactions with age are interesting and
because it is a factor, it is easier to handle than a covariate where age was entered as a
factor. The number of animals in the four age classes is 25, 44, 24, and 32, respectively. The age class is evenly distributed in the combinations of mating period and
treatment groups. Ewes were slaughtered at 75–80 days after the last mating, and the
number of ovulations and number of fetuses were determined. Ovulation numbers
ranged from 1 to 5. For six animals, the number of ovulations was not known, so
these ewes were excluded from the database.
(a) Analyze
the dataset
using a GLMM
with the predictor:
ηijkl = η + τi + αj + βk + (τα)ij + (τβ)ik + (ταβ)ijk + bl, where τ, α, and β are the
fixed effects of treatment, age, and mating period and b is the random effect due
to animal. Assuming that each b has normal distribution with a zero mean and
variance σ 2b , and under the assumption that the number of ovulations and the
number of fetuses have a Poisson distribution.
(b) From the analyses performed, do you observe the presence of overdispersion in
the dataset? If so, propose an alternative distribution for the analysis for this
dataset.
(c) Reanalyze the dataset using the same model as before with the new data
distribution.
(d) Compare and contrast the results of these analyses.
(e) Summarize the relevant results.
Exercise 5.3.5 The following example deals with one of the most harmful insects in
the root system of the main crops, whose common name is “blind hen.” The
experiment consisted of six treatments formulated for larval control in a randomized
block arrangement (A, B, C, D, E, and F). The count per area shows the number of
larvae in two age groups (a and b) (Table 5.49).
T
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
A
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
M
1
1
1
1
1
1
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
2
N
2
2
2
1
2
2
2
2
1
2
2
2
1
2
2
2
4
3
2
2
2
2
3
2
x
1
2
2
1
1
2
1
1
1
1
1
1
1
1
2
2
2
2
2
2
1
2
1
2
T
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
A
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
4
4
M
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
2
2
2
2
2
2
1
1
n
2
2
2
3
2
3
3
3
3
2
3
3
2
3
2
2
4
4
4
3
3
2
3
2
x
1
2
1
2
2
3
2
1
2
1
3
2
1
3
2
1
4
3
1
3
2
2
2
1
T
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
A
4
4
4
4
4
4
4
4
4
4
4
1
1
1
1
1
1
1
1
1
1
1
2
2
M
1
1
1
1
2
2
2
2
2
2
2
1
1
1
1
1
2
2
2
2
2
2
1
1
n
3
3
2
3
2
4
2
4
2
2
5
1
2
2
1
1
1
1
1
1
1
1
2
1
X
2
3
2
3
2
4
2
3
2
2
2
1
1
1
1
1
1
1
1
1
1
1
2
1
T
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
A
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
M
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
n
3
2
1
2
2
2
2
3
2
2
2
2
2
2
2
2
1
2
1
1
2
3
2
2
x
2
1
1
1
2
1
2
2
2
1
2
2
2
2
2
2
1
2
1
1
2
3
1
2
T
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
A
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
M
1
1
2
2
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
n
3
2
2
2
2
2
2
2
2
2
2
2
2
3
2
2
2
2
2
2
2
3
3
x
2
2
1
2
1
2
2
2
1
2
2
2
2
3
2
2
2
2
1
2
2
3
3
Table 5.48 Factors T (Treatment: 1: Fecundin; 2: Control), A (Age: 1: ≤0.5; 2: 0.5 < - ≤ 1.5; 3: 1.5 < - ≤ 2.5; 4:≥2.5 years); M (Mating period: 1: October
1; 2: October 22); n (number of ovulations), and x (number of fetuses)
5.3
Exercises
177
178
5 Generalized Linear Mixed Models for Counts
Table 5.49 Results of the
blind hen experiment
Trt
1
2
3
4
5
A
A
5
4
4
1
2
b
7
2
4
5
2
B
A
5
12
1
5
3
B
14
5
14
9
8
C
A
0
2
2
2
0
B
3
3
2
7
0
D
A
1
1
1
3
5
b
7
6
7
7
4
E
A
1
3
1
0
1
b
10
5
8
3
6
F
A
4
4
7
3
1
b
13
11
10
12
8
(a) Write down the analysis of variance table (sources of variation and degrees of
freedom).
(b) Write down the components of the GLMM.
(c) Analyze the dataset with the model proposed in (b).
(d) Does the proposed model in (b) adequately describe the variation observed in the
dataset? Summarize the relevant results.
Appendix 1
Data: Subcultures
sub1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
Rep1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
2
3
4
NB
18
16
15
15
11
17
10
8
17
13
16
15
12
15
8
8
15
15
14
8
15
11
12
18
sub1
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
Rep1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
NB
24
24
19
25
24
20
24
20
19
26
22
23
24
23
24
28
29
34
24
24
25
28
24
32
sub1
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
Rep1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
5
6
7
8
NB
45
44
45
44
52
47
46
45
48
56
54
44
54
62
55
45
56
62
45
45
46
48
55
45
sub1
8
8
8
8
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
9
9
Rep1
9
10
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
11
12
NB
53
59
57
65
63
55
50
52
55
50
53
52
48
44
54
55
51
58
47
42
50
48
48
53
(continued)
Appendix 1
sub1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
Rep1
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
1
179
NB
8
17
8
18
22
19
19
24
12
12
11
21
10
15
20
22
20
13
18
19
sub1
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
Rep1
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
NB
34
30
26
27
29
38
38
37
41
46
44
54
45
60
57
51
54
51
62
53
sub1
7
7
7
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
Rep1
9
10
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
NB
45
44
52
45
43
58
62
45
63
56
55
50
53
58
56
50
57
60
50
52
sub1
9
9
9
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Rep1
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
NB
54
59
58
46
38
29
30
31
33
35
59
37
44
42
41
45
38
40
Data: Beatles
Row
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
Column
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
Treatment
S
U
U
TR
S
TR
TR
S
TR
U
U
S
U
TR
U
S
S
TR
U
TR
Count
3
6
2
7
1
5
5
4
5
8
6
3
3
6
4
3
4
7
3
4
(continued)
180
Row
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
5
Column
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
Generalized Linear Mixed Models for Counts
Treatment
TR
S
U
S
TR
S
S
U
TR
U
S
U
S
TR
TR
Count
5
2
3
6
8
5
6
7
9
4
6
5
6
9
9
B
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
Count
14
7
5
7
5
14
5
9
0
0
10
13
20
53
21
7
12
31
32
22
49
16
7
14
20
Data: Weed counts
Block
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
A
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
(continued)
Appendix 1
Block
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
181
A
7
7
7
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
1
1
1
1
2
2
2
2
3
3
3
3
B
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Count
20
16
6
9
9
9
19
31
11
30
29
25
11
15
23
7
22
20
3
0
28
18
18
55
58
18
19
14
44
19
17
12
8
44
0
29
11
5
49
99
66
11
15
(continued)
182
Block
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
A
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
Generalized Linear Mixed Models for Counts
B
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Count
9
8
9
21
49
49
17
22
41
21
48
11
58
34
28
20
6
9
20
0
10
0
7
9
9
29
22
4
22
31
32
41
112
44
24
28
8
8
11
10
117
78
36
38
Shade
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Coffee data
Clone
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
Tray
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
Rep
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R2
R2
R2
R2
R2
R2
R2
y1
2
2
2
0
0
2
2
0
2
2
0
2
2
2
0
2
2
2
2
2
2
0
0
2
2
y2
2
2
0
2
0
0
0
2
2
2
2
2
1
2
2
2
2
4
2
2
2
2
2
2
0
y3
4
4
4
5
3
2
4
6
4
6
4
4
5
4
6
6
4
6
5
4
3
4
4
6
4
y4
4
4
5
2
2
2
5
6
5
5
4
6
7
5
8
5
5
7
6
5
5
6
6
5
6
y5
6
2
8
6
6
6
8
8
6
8
8
8
8
6
10
8
8
8
8
8
6
8
8
2
8
y6
4
2
7
3
6
6
3
10
7
9
8
6
6
4
10
7
8
10
9
10
8
10
10
8
10
y7
5
4
5
9
7
6
3
10
7
7
4
6
4
7
8
6
5
11
6
6
5
6
7
4
9
y8
5
2
4
8
6
5
6
10
5
4
4
4
4
7
8
2
3
11
6
4
4
7
4
5
8
y9
5
4
4
8
7
5
2
9
5
3
4
4
2
8
7
2
2
10
6
2
4
7
3
4
6
y10
4
8
6
9
5
10
11
5
5
10
5
4
11
6
4
6
9
11
8
2
6
9
7
4
13
(continued)
7
8
7
4
13
y11
7
12
5
8
6
14
9
9
7
6
7
8
13
11
2
12
12
13
9
Appendix 1
183
Shade
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Clone
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
Coffee data (continued)
Tray
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
Rep
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
y1
0
0
2
2
0
2
2
2
2
0
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
y2
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
2
0
2
0
2
2
2
4
4
y3
4
5
4
6
5
6
4
4
4
4
4
6
4
6
4
2
2
4
4
2
4
4
4
6
4
y4
4
4
6
6
6
6
4
6
6
6
6
6
6
4
6
4
2
4
6
4
4
6
5
5
6
y5
8
8
6
6
8
10
8
8
8
8
8
6
6
6
8
4
2
6
8
6
6
8
6
4
8
y6
10
10
10
6
10
8
10
10
8
10
8
8
8
8
10
4
4
6
6
6
4
8
6
6
10
3
2
4
2
7
5
2
9
y7
10
8
3
4
6
5
8
6
7
10
8
5
8
10
4
2
y8
3
4
2
3
3
4
10
3
8
5
4
7
1
8
3
6
2
2
8
2
2
7
2
2
2
y10
5
11
3
7
1
1
11
9
13
11
14
5
11
8
10
6
8
4
8
7
1
8
12
8
y9
3
2
2
7
3
3
12
3
12
5
4
1
7
7
6
4
2
1
3
2
4
4
6
4
6
10
6
12
9
2
8
12
12
y11
15
5
8
7
10
9
9
6
12
11
11
12
9
9
11
184
5 Generalized Linear Mixed Models for Counts
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH2
CH2
CH2
CH2
CH2
R3
R3
R3
R3
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R1
R1
R1
R1
R1
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
2
2
2
0
0
2
2
2
2
2
2
2
2
2
3
2
2
2
2
0
2
0
0
0
0
0
2
4
0
0
4
2
4
2
2
2
2
2
6
4
6
6
4
6
6
2
2
2
3
4
2
4
2
4
4
4
5
6
4
6
3
4
4
4
4
6
5
6
6
5
6
6
2
2
2
2
4
4
5
2
5
5
4
6
5
5
6
5
6
6
4
4
8
4
8
8
8
10
8
4
4
2
4
8
6
4
2
6
10
4
8
4
8
8
6
10
8
8
6
10
10
8
8
8
8
8
6
3
4
4
10
7
2
4
8
10
6
10
8
8
10
8
10
10
8
6
12
10
6
6
2
10
12
6
3
6
3
4
7
1
5
8
6
3
9
2
7
8
5
11
11
2
3
9
2
3
5
3
2
3
1
2
6
1
8
6
1
2
6
5
3
4
3
4
3
10
11
10
1
5
3
8
1
4
4
1
6
9
10
1
3
11
3
16
10
9
13
7
1
6
6
1
6
7
2
2
5
7
2
3
4
3
4
9
11
1
5
11
11
12
18
2
11
2
10
4
3
5
3
2
2
(continued)
3
9
2
6
5
1
7
9
5
1
2
11
6
16
14
8
14
11
6
11
16
16
23
4
11
10
Appendix 1
185
Shade
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Clone
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
Coffee data (continued)
Tray
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
Rep
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
y1
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
y2
2
2
2
2
0
2
2
2
2
2
2
2
4
0
0
2
2
2
0
2
0
2
2
0
2
y3
6
4
6
4
4
4
4
6
6
6
5
6
4
4
4
6
4
4
2
6
2
4
4
4
4
y4
6
6
6
6
4
6
6
6
6
6
6
6
7
6
6
6
6
6
4
4
4
6
4
4
4
y5
8
8
10
8
8
8
8
10
10
8
8
8
10
8
8
8
6
8
6
6
4
6
8
8
8
y6
8
10
10
10
10
10
10
10
10
8
10
12
11
8
8
10
10
10
8
10
8
10
10
8
8
y7
9
7
11
9
6
12
10
9
12
10
10
6
10
4
5
10
4
12
10
10
4
10
4
2
8
y9
2
1
5
6
5
14
9
4
9
5
9
7
8
5
6
3
2
2
2
9
3
y8
2
1
6
6
6
12
10
5
9
8
12
5
9
7
6
3
2
2
2
5
8
4
11
5
9
3
10
3
8
1
10
7
7
9
8
12
6
9
14
11
9
15
16
10
7
8
14
9
10
10
7
9
8
5
14
y11
1
y10
2
1
7
186
5 Generalized Linear Mixed Models for Counts
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
R2
R2
R2
R2
R2
R2
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R4
R4
R4
0
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
3
2
2
2
2
2
2
0
2
2
2
2
2
2
2
0
2
3
3
2
0
2
2
4
6
6
6
6
4
6
4
4
6
4
4
4
6
6
4
6
6
6
4
6
5
5
6
4
4
4
6
6
6
6
5
6
6
6
5
6
5
6
6
6
6
6
5
6
5
4
8
6
6
6
5
6
6
8
8
10
8
8
8
10
10
8
8
8
8
8
10
10
8
8
8
8
8
8
10
10
10
4
8
8
8
10
10
12
10
10
10
10
10
10
12
10
12
10
10
8
6
8
10
10
10
12
12
10
8
10
10
6
10
7
4
8
10
4
2
6
3
10
8
10
10
10
4
6
4
6
6
6
11
6
5
10
8
12
1
1
1
2
1
5
1
1
1
3
2
3
7
1
7
2
5
12
9
1
2
2
1
3
3
4
7
1
6
2
6
8
1
12
1
3
11
12
12
5
6
14
8
7
3
13
4
3
5
5
5
8
5
1
6
4
6
10
6
1
5
4
2
1
3
1
(continued)
10
2
2
11
5
14
6
9
18
11
8
4
4
14
6
4
11
8
Appendix 1
187
Shade
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Clone
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
Coffee data (continued)
Tray
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
Rep
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
y1
2
2
0
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
y2
2
0
2
2
0
2
2
2
0
2
0
2
3
0
3
0
2
2
0
0
2
2
0
2
2
y3
6
5
4
6
4
6
6
6
4
6
4
6
4
4
7
4
5
2
4
5
3
4
4
4
4
y4
6
6
4
6
6
6
6
6
6
6
4
8
5
6
7
6
6
4
6
5
5
6
4
6
4
y5
8
8
6
8
8
10
10
8
8
10
8
8
8
8
10
7
8
6
6
6
7
9
8
8
8
y6
8
8
8
8
10
9
8
10
10
8
8
10
8
10
10
10
8
8
8
8
8
8
8
10
8
y7
8
5
8
8
12
11
9
10
9
8
6
10
8
9
9
9
10
7
8
10
8
10
5
9
9
y8
4
4
6
3
9
9
8
7
9
7
2
7
5
7
10
7
6
2
9
4
9
10
5
7
7
y9
3
6
6
3
9
9
7
6
6
1
1
8
5
7
10
4
7
3
9
4
9
6
3
9
6
y10
7
10
4
10
7
8
5
7
4
7
3
7
3
6
12
4
7
5
9
2
9
5
3
11
7
y11
13
9
4
6
9
18
10
12
8
10
4
7
11
14
20
13
4
7
8
2
10
8
10
14
5
188
5 Generalized Linear Mixed Models for Counts
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
R1
R1
R1
R1
R1
R1
R1
R1
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R3
2
2
2
2
0
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
0
2
2
2
0
2
2
2
0
0
0
2
2
0
2
1
2
2
2
2
2
2
2
2
0
2
2
5
4
4
4
6
6
4
4
4
4
4
4
2
4
4
6
6
4
6
4
6
4
6
4
4
4
5
4
6
6
4
6
6
6
6
6
6
6
6
2
4
6
6
8
6
6
6
6
6
6
6
6
6
6
9
8
8
6
8
8
9
8
8
8
8
8
6
8
10
10
10
8
9
8
10
8
8
8
8
8
8
8
8
10
8
10
10
10
10
8
8
10
10
8
8
8
10
10
8
8
8
10
8
10
10
10
10
10
7
10
9
5
5
10
8
6
10
3
12
12
8
6
8
10
9
7
2
8
12
8
10
10
8
10
10
5
8
8
12
10
10
9
2
6
8
10
11
9
9
10
9
15
5
14
12
10
12
5
17
12
3
6
1
4
10
14
12
8
3
8
4
4
9
7
11
9
3
11
8
3
10
2
5
7
9
1
12
11
8
7
6
5
4
6
7
4
5
10
7
11
(continued)
12
14
17
20
8
10
15
5
22
18
6
10
7
10
12
17
9
8
3
9
Appendix 1
189
Shade
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Clone
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
Coffee data (continued)
Tray
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
Rep
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R4
R4
R4
R4
R4
R4
R4
R4
y1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
y2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
2
2
2
y3
4
5
6
4
3
5
6
4
4
4
4
4
4
5
4
4
4
6
4
4
4
4
6
6
5
y4
5
5
4
4
4
6
6
6
5
6
6
6
6
5
6
6
6
6
5
6
6
6
6
6
5
y5
9
9
9
8
6
8
8
6
7
8
8
8
8
8
8
8
8
8
8
8
8
8
7
8
7
y6
10
12
8
8
8
10
11
12
8
10
6
10
10
10
8
8
10
8
12
8
10
9
10
10
8
y7
10
6
6
10
8
11
4
7
4
5
6
4
12
4
3
5
6
8
6
10
12
11
6
11
5
13
6
4
3
1
12
13
8
8
3
2
2
8
5
10
2
3
4
10
3
9
10
4
7
9
8
1
2
3
6
1
8
6
1
6
6
2
7
y9
5
2
1
9
y8
6
2
14
14
6
7
8
18
6
22
23
13
16
2
5
17
4
12
11
11
15
y11
18
8
1
10
6
3
7
6
16
10
1
2
2
10
11
3
1
9
y10
3
190
5 Generalized Linear Mixed Models for Counts
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Roja
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
2
0
2
2
0
0
0
0
0
2
0
2
2
0
0
3
2
2
2
2
0
2
6
4
6
6
6
6
6
4
4
5
4
4
2
2
6
3
4
4
4
2
4
4
4
4
4
6
4
6
7
8
8
8
8
8
5
6
6
3
5
2
2
4
2
5
2
5
2
3
5
6
4
5
4
6
8
8
10
9
10
10
8
8
8
8
2
7
2
4
4
4
8
3
7
2
6
7
5
7
7
6
8
10
12
10
10
8
10
10
10
10
6
2
5
4
5
4
4
6
2
4
7
3
6
6
8
9
4
6
2
9
4
2
1
3
3
2
7
4
2
2
2
6
2
1
3
4
3
3
4
10
4
5
3
9
5
10
10
10
9
10
6
12
8
5
7
10
12
12
8
1
1
1
7
1
3
1
2
10
10
9
10
11
7
11
4
4
4
6
4
5
4
2
(continued)
8
12
14
14
6
6
1
2
9
14
16
24
23
13
2
14
12
14
21
11
9
1
13
11
14
7
Appendix 1
191
Shade
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Clone
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
Coffee data (continued)
Tray
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
Rep
R1
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R3
R3
R3
R3
R3
R3
y1
2
2
2
2
0
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
y2
2
2
2
2
0
0
0
2
2
0
2
0
2
2
2
0
3
2
2
2
2
0
2
2
2
y3
6
4
4
4
3
2
2
4
4
4
4
4
4
4
6
5
4
4
5
6
2
4
6
4
4
y4
6
3
6
3
3
2
1
4
4
3
6
4
1
4
6
3
4
5
6
6
2
4
7
6
4
y5
7
2
8
4
4
2
2
4
8
4
6
5
4
4
6
4
5
5
8
8
4
4
4
8
6
y6
3
8
8
2
6
8
2
2
6
4
7
4
6
5
7
5
5
7
4
7
5
6
2
8
4
y7
3
6
8
2
8
5
3
3
4
2
1
4
3
4
5
4
2
5
4
4
2
1
9
3
4
7
4
1
2
9
6
3
8
1
1
1
6
5
5
8
3
2
3
3
2
5
y10
3
10
7
3
2
1
2
3
2
4
2
2
1
2
1
1
2
2
11
1
2
5
3
2
1
2
y9
3
2
y8
3
1
7
7
12
16
2
5
9
6
5
11
11
2
4
5
5
5
y11
2
10
192
5 Generalized Linear Mixed Models for Counts
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
2
2
2
2
2
0
4
2
4
4
2
0
0
0
0
2
0
0
0
0
0
2
4
2
2
3
4
4
6
4
4
4
4
4
6
3
6
4
4
4
2
4
4
4
2
3
2
3
4
10
4
4
3
6
5
5
7
8
6
6
5
7
6
5
4
3
3
2
4
4
4
2
3
2
4
4
7
4
5
3
6
8
9
7
6
7
8
7
5
7
5
4
5
6
2
6
7
7
4
3
2
6
7
9
7
7
2
5
4
4
5
5
9
5
3
4
7
4
4
3
4
2
5
4
6
2
4
4
3
3
8
8
7
2
8
4
3
1
8
6
5
6
1
6
1
7
3
3
5
5
2
1
2
3
8
2
1
2
4
1
1
5
2
1
3
3
4
3
5
3
6
2
5
4
5
3
6
3
4
2
5
5
3
2
8
3
8
9
4
2
10
4
9
2
5
5
3
5
6
6
(continued)
4
9
3
13
12
4
2
11
10
9
8
2
12
10
7
6
Appendix 1
193
Shade
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Clone
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
Coffee data (continued)
Tray
CH1
CH1
CH1
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
Rep
R4
R4
R4
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R2
R2
R2
R2
y1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
y2
0
0
2
2
2
2
2
0
0
2
2
2
0
2
2
0
2
0
2
2
2
2
2
2
0
y3
4
4
6
4
4
4
6
4
4
4
4
4
4
2
6
4
4
6
4
4
4
6
4
4
4
y4
5
4
6
6
6
4
6
4
4
6
6
4
4
2
6
6
6
6
4
6
6
6
6
6
6
y5
6
6
8
8
8
8
10
8
7
8
8
8
8
4
10
8
5
10
6
8
10
10
10
10
8
y6
4
4
2
9
8
8
10
8
6
6
8
8
8
5
10
10
7
8
9
4
7
8
5
8
8
10
6
12
4
1
11
2
6
11
10
3
5
8
9
9
5
9
4
9
8
6
y7
3
2
1
2
6
4
1
2
10
3
1
1
1
6
1
1
1
10
12
11
10
4
12
9
9
4
10
y9
2
1
3
12
y8
2
1
4
2
11
9
13
10
2
1
1
12
15
16
12
11
9
y11
10
4
4
12
5
7
9
7
14
15
12
12
5
y10
1
1
2
12
194
5 Generalized Linear Mixed Models for Counts
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
0
2
2
2
2
2
0
1
2
2
1
0
2
5
0
2
2
0
0
2
2
0
0
2
2
0
0
4
4
4
6
4
6
6
4
4
4
4
4
4
6
4
4
6
4
4
4
4
4
4
4
4
4
4
6
6
6
6
6
6
6
7
5
6
4
6
6
9
6
6
6
3
4
6
4
4
4
8
8
6
4
7
8
10
10
8
9
10
8
10
10
8
10
10
11
8
8
8
7
8
8
8
7
8
8
8
8
8
9
10
8
10
8
9
10
10
8
6
8
9
4
10
7
8
8
7
7
6
8
5
8
7
8
4
8
3
6
5
6
6
8
6
4
6
5
1
6
1
3
5
5
4
3
2
4
2
2
1
5
7
8
9
8
9
8
10
7
10
5
9
6
3
6
5
5
6
2
5
9
8
9
1
3
10
2
1
1
5
3
2
5
1
1
3
2
2
7
6
4
5
7
8
7
4
7
5
3
5
(continued)
6
6
11
6
6
9
9
6
8
4
3
1
3
6
7
6
11
3
10
11
9
9
7
12
4
7
5
6
3
8
9
6
7
5
9
Appendix 1
195
Shade
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Clone
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
Coffee data (continued)
Tray
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH3
CH3
Rep
R3
R3
R3
R3
R3
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R1
R1
y1
2
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
y2
0
0
2
2
2
0
2
2
0
0
0
0
2
2
2
2
2
0
2
2
2
2
4
0
2
y3
2
4
4
4
4
4
4
4
4
4
4
2
2
4
4
4
4
4
4
4
4
4
6
4
2
y4
2
6
7
6
8
5
6
7
6
6
6
4
4
6
4
6
6
6
6
5
6
6
8
6
6
y5
6
8
12
8
8
2
10
5
5
8
7
4
6
10
5
8
7
9
2
6
8
8
10
8
8
y6
9
10
9
6
4
7
9
3
6
6
5
7
8
7
5
8
6
5
4
8
8
5
8
8
7
8
4
7
5
7
9
6
3
7
8
5
9
9
9
13
7
6
7
7
7
8
10
10
12
6
y11
7
2
12
10
4
5
2
4
5
6
4
2
1
1
1
1
2
3
y10
7
1
8
7
y9
6
1
3
4
3
2
3
2
1
9
1
7
1
7
8
4
5
y8
5
2
4
1
5
5
2
1
1
y7
9
9
4
5
1
5
3
1
1
2
196
5 Generalized Linear Mixed Models for Counts
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
2
2
2
2
0
0
0
2
2
0
2
2
2
2
2
2
2
0
4
0
0
2
0
0
2
0
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
2
4
4
4
4
3
4
4
4
4
4
4
6
4
6
6
6
6
6
6
4
4
6
6
6
6
6
6
6
6
4
6
4
4
6
4
4
4
4
7
8
8
8
8
8
8
8
8
6
10
10
10
8
8
8
8
9
8
10
7
8
8
8
8
8
8
10
6
8
5
10
9
9
6
8
5
7
10
8
8
10
8
10
5
8
8
7
8
8
8
8
7
8
9
10
9
9
11
10
11
8
8
6
11
12
9
6
12
9
12
5
10
5
7
10
10
10
10
9
10
12
9
9
8
3
10
7
5
6
4
12
11
5
10
7
3
12
8
10
9
7
9
10
6
9
8
10
3
8
11
4
11
6
3
11
8
8
8
5
10
10
6
9
10
10
10
11
10
9
3
8
7
5
11
11
11
10
2
8
8
6
2
4
10
11
10
11
14
3
10
9
10
12
6
10
12
9
10
14
9
(continued)
16
13
11
11
4
10
12
8
8
12
16
11
11
17
20
5
14
14
16
16
9
15
14
10
16
9
6
Appendix 1
197
Shade
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Clone
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
Coffee data (continued)
Tray
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
Rep
R2
R2
R2
R2
R2
R2
R2
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
y1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
y2
2
0
0
0
0
2
0
2
0
2
0
0
0
2
2
0
2
0
0
0
0
2
4
2
2
y3
4
4
4
4
4
4
4
4
4
4
2
2
2
4
4
4
4
4
4
2
4
4
6
4
4
y4
6
6
6
6
6
6
6
6
6
6
2
2
2
6
6
6
4
6
4
4
8
6
6
6
6
y5
8
8
8
8
8
8
8
8
8
8
2
4
4
10
8
10
8
8
10
8
10
8
8
8
8
y6
10
10
10
8
8
10
7
6
7
8
3
6
6
8
5
8
8
8
7
6
10
8
8
10
9
y7
12
4
12
10
8
10
5
4
2
8
3
8
8
10
9
7
10
10
9
8
12
7
9
8
8
y8
9
7
5
5
6
12
12
5
2
6
3
4
7
8
3
2
6
6
6
7
11
10
7
5
10
6
4
2
1
8
10
9
11
14
12
11
5
9
11
4
6
5
10
13
19
12
16
18
26
15
11
16
8
21
22
18
12
14
4
5
6
7
9
y11
y10
7
6
7
3
7
13
14
6
9
5
5
4
5
8
y9
9
3
4
4
5
12
4
2
4
8
3
4
8
8
1
1
2
198
5 Generalized Linear Mixed Models for Counts
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Perla
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R1
R1
R1
R1
R1
R1
R1
R1
R1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
2
2
2
2
0
0
2
0
0
2
0
0
2
2
0
2
2
2
2
0
4
2
2
0
0
0
0
2
2
0
4
4
6
3
5
4
4
4
4
4
4
6
6
5
4
4
5
3
4
6
4
2
2
2
4
4
2
6
6
6
5
6
4
6
6
6
4
6
6
4
5
4
6
5
5
6
6
6
4
2
2
4
5
3
7
10
8
7
6
6
8
8
6
6
8
8
9
8
6
8
7
6
6
6
8
6
2
2
6
6
4
5
7
5
8
8
2
8
7
8
4
4
6
5
4
5
6
5
4
8
8
8
8
4
4
8
8
6
1
5
1
2
5
5
6
8
3
4
3
4
4
9
7
5
3
1
2
5
5
6
6
10
4
1
1
1
1
5
3
1
5
4
4
5
5
2
5
9
6
4
9
5
10
2
2
4
4
4
1
2
4
4
7
1
1
1
1
5
5
5
5
7
5
4
6
3
9
2
7
16
4
10
4
5
3
7
8
3
11
5
(continued)
6
4
7
10
9
11
5
8
20
8
14
11
8
7
8
8
7
14
6
Appendix 1
199
Shade
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Clone
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
Coffee data (continued)
Tray
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
Rep
R1
R1
R1
R1
R1
R1
R1
R1
R1
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
y1
2
2
2
2
2
0
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
2
2
2
y2
0
0
2
2
2
0
2
2
2
2
2
0
0
2
2
2
0
2
2
0
0
0
2
2
2
y3
4
4
4
4
4
2
6
4
6
6
4
4
4
4
4
6
2
6
4
2
4
4
6
6
4
y4
6
4
6
6
4
4
6
4
6
6
6
6
6
4
6
6
2
6
4
2
2
4
6
6
6
y5
6
6
8
8
6
6
8
6
8
8
6
6
4
6
8
4
6
6
6
4
2
6
6
8
6
y6
4
6
8
8
8
8
8
6
8
8
8
6
4
8
8
4
6
8
4
6
2
6
8
7
8
6
3
6
4
6
2
4
4
6
2
4
4
11
3
1
1
3
2
3
3
3
6
4
7
10
9
7
3
5
4
5
y10
8
6
8
8
3
y9
4
y8
4
y7
4
13
2
2
13
22
16
9
8
y11
200
5 Generalized Linear Mixed Models for Counts
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
R2
R2
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R4
R4
R4
R4
R4
R4
R4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
2
2
0
0
0
0
2
2
2
2
2
2
2
2
2
2
4
4
2
0
2
2
2
2
2
6
6
6
6
4
2
2
4
6
6
4
4
6
6
6
6
6
6
6
6
6
4
6
4
6
4
4
6
7
6
6
6
6
2
4
6
6
6
6
6
4
6
6
6
6
6
8
6
6
6
5
6
6
6
8
8
8
8
8
8
2
4
8
8
8
8
8
6
8
8
8
8
8
8
8
8
8
8
8
8
8
6
4
10
10
10
4
2
8
6
10
6
6
10
4
6
10
9
10
8
8
6
4
4
8
6
6
6
3
6
2
5
5
4
6
2
8
8
6
4
5
5
1
6
4
4
3
4
5
12
7
6
3
4
2
1
2
4
4
6
6
2
4
3
4
3
2
1
3
2
2
3
2
9
5
6
1
2
5
2
13
1
7
10
13
18
3
3
7
1
7
7
1
9
4
7
4
1
2
1
5
1
6
6
1
3
1
4
5
9
6
4
(continued)
11
2
20
2
10
18
17
18
10
9
12
4
10
9
15
14
8
10
Appendix 1
201
Shade
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Clone
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
Coffee data (continued)
Tray
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH1
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
Rep
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
y1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
2
0
2
2
2
2
2
2
2
2
y2
0
0
2
0
2
0
2
2
4
2
2
2
0
2
0
2
2
0
0
2
2
2
0
2
2
y3
4
4
6
2
4
6
6
6
6
6
6
4
4
6
2
4
2
6
4
4
4
6
4
6
4
y4
6
4
6
2
6
4
5
6
6
5
4
6
6
6
4
4
2
6
6
6
6
6
10
6
7
y5
6
8
8
2
8
6
8
8
10
8
4
8
8
8
6
6
4
8
8
8
8
8
8
8
10
y6
2
2
3
4
6
6
8
6
10
6
4
8
7
8
8
10
6
10
8
10
10
10
7
10
10
4
3
4
5
1
5
4
3
1
7
5
4
7
8
8
3
5
6
6
4
10
9
9
8
6
8
5
6
7
5
10
10
9
5
y8
5
3
2
6
y7
9
3
3
4
3
7
1
5
4
3
4
4
1
6
3
3
1
2
3
2
y9
5
3
2
6
2
9
4
1
y10
2
5
3
2
9
8
5
7
11
8
10
7
4
1
2
9
1
3
8
5
7
y11
2
6
4
5
8
15
11
10
16
11
14
10
8
5
5
9
12
202
5 Generalized Linear Mixed Models for Counts
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
R1
R1
R1
R1
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R3
R3
R3
R3
R3
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
3
2
0
2
2
2
2
2
2
0
2
0
0
2
2
2
2
2
0
0
0
2
2
0
0
4
4
6
6
4
6
4
6
4
4
4
4
4
4
6
6
6
6
4
4
4
4
6
6
4
4
4
8
6
6
6
6
6
6
6
4
4
6
4
6
4
6
6
6
6
8
6
6
6
6
6
6
6
6
10
8
8
8
8
8
6
8
8
6
8
6
8
8
8
10
8
8
8
8
8
8
8
10
8
8
8
10
6
6
6
7
9
8
5
7
8
6
8
8
9
7
10
10
10
10
10
10
7
8
4
8
8
6
11
3
6
7
10
10
2
6
7
5
4
6
8
8
6
8
9
5
3
10
10
10
9
7
9
12
3
5
2
2
7
7
5
5
2
6
7
10
6
8
7
10
10
7
10
9
4
2
8
6
4
1
3
8
2
2
6
7
6
4
3
3
8
2
2
8
9
2
8
6
10
(continued)
12
14
12
20
4
6
7
11
10
4
2
7
7
7
10
2
10
11
12
8
2
14
6
9
2
2
3
2
7
Appendix 1
203
Shade
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Clone
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
Coffee data (continued)
Tray
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
CH2
Rep
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
y1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
y2
2
2
2
2
2
2
2
2
2
2
0
0
0
2
2
2
0
0
2
2
2
0
2
0
2
y3
4
6
4
4
6
6
4
4
4
6
4
4
6
6
6
6
2
4
6
6
6
4
6
6
6
y4
6
6
4
6
6
6
6
6
4
8
6
6
8
6
6
6
4
4
6
6
6
4
6
6
4
y5
6
8
6
8
8
8
6
8
8
8
8
8
8
8
8
8
6
4
8
8
8
6
6
8
8
y6
5
10
4
6
4
10
6
10
8
12
10
10
8
8
8
3
5
2
2
8
6
2
8
10
4
4
3
4
5
1
3
4
2
4
3
4
6
7
5
1
2
2
5
8
9
7
8
2
2
9
10
13
10
6
7
5
4
2
2
9
1
5
4
5
4
7
13
14
18
12
7
11
14
12
6
5
12
8
4
5
2
3
7
8
11
6
9
y11
y10
y9
2
3
2
4
4
y8
4
4
5
4
6
y7
7
9
5
8
7
7
7
11
8
8
12
8
9
204
5 Generalized Linear Mixed Models for Counts
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
CH2
CH2
CH2
CH2
CH2
CH2
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
R4
R4
R4
R4
R4
R4
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R1
R2
R2
R2
2
2
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
2
0
2
2
0
2
2
2
2
0
2
2
0
4
2
2
2
2
2
2
2
0
0
0
2
6
4
6
4
4
4
6
6
4
6
6
2
6
4
4
4
4
4
6
6
6
6
6
4
4
4
6
6
8
6
6
6
6
6
6
4
4
6
4
6
6
6
6
4
6
6
6
8
8
6
6
6
6
6
10
10
8
8
8
8
8
8
4
6
8
2
8
6
8
8
6
7
8
8
8
6
8
8
8
8
10
10
6
8
8
8
10
9
6
4
8
8
6
10
8
10
10
8
6
10
9
5
8
10
10
10
6
6
2
2
5
5
7
5
6
10
12
6
4
8
8
9
6
9
9
9
6
9
7
8
8
6
7
10
12
8
3
7
10
8
6
8
8
8
8
7
7
5
2
4
5
4
3
10
1
8
6
5
3
6
4
10
12
1
2
7
11
7
11
4
9
10
3
10
1
1
2
9
7
7
7
9
10
3
4
3
6
2
5
1
5
6
9
9
6
3
9
10
6
4
4
5
(continued)
2
6
11
3
9
12
9
10
7
6
4
10
15
5
7
16
6
9
8
8
8
Appendix 1
205
Shade
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Clone
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
Coffee data (continued)
Tray
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
Rep
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R2
R3
R3
R3
R3
R3
R3
R3
R3
R3
R3
y1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
y2
0
2
0
2
2
2
0
2
2
2
2
2
2
0
0
2
0
2
0
0
2
0
0
0
0
y3
2
6
4
6
6
6
6
4
6
6
6
6
4
4
4
6
4
4
4
4
6
4
4
4
4
y4
2
6
4
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
y5
4
6
6
8
8
8
8
6
6
10
8
8
8
8
6
8
8
8
8
8
10
7
7
8
8
y6
6
6
7
10
6
7
7
4
5
7
7
8
5
2
7
6
8
4
9
8
8
8
8
5
8
y7
6
7
6
10
7
6
7
4
7
6
8
9
6
4
6
2
4
5
4
5
9
4
8
3
9
y9
2
4
3
10
5
5
2
3
6
6
4
6
5
6
4
4
10
2
7
6
y8
2
4
5
12
6
7
3
6
5
3
5
4
4
6
4
4
11
2
6
7
2
6
4
7
7
6
7
4
4
2
7
4
5
3
4
7
4
8
14
2
2
2
11
8
5
2
2
1
9
10
4
1
5
6
3
6
y11
y10
206
5 Generalized Linear Mixed Models for Counts
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
Negra
C4
C4
C5
C5
C5
pf
pf
pf
C1
C1
C1
C2
C2
C2
C3
C3
C3
C4
C4
C4
C5
C5
C5
pf
pf
pf
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
CH3
R3
R3
R3
R3
R3
R3
R3
R3
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
R4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
0
0
0
2
0
0
2
0
0
0
0
2
0
2
2
4
4
6
2
4
4
6
4
4
4
4
4
4
4
2
2
4
6
4
6
4
4
6
4
4
4
4
6
6
6
4
6
6
6
8
8
4
6
6
4
6
2
4
6
4
6
4
4
8
6
6
6
8
8
8
8
8
8
6
8
8
8
8
8
8
6
6
6
6
10
6
8
6
8
8
8
8
8
8
10
7
8
3
6
3
5
8
5
8
7
8
5
7
5
6
10
7
6
8
7
6
8
8
10
8
10
8
8
2
5
4
5
4
3
9
9
1
6
7
7
6
10
7
8
10
6
4
8
8
11
3
5
4
7
2
3
6
7
4
1
2
10
6
2
5
4
8
1
8
1
3
2
3
5
4
4
2
2
6
8
4
2
1
6
7
6
8
2
7
2
2
1
2
7
2
8
10
6
2
2
5
8
6
2
1
10
7
4
9
2
9
Appendix 1
207
208
5
Generalized Linear Mixed Models for Counts
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give appropriate
credit to the original author(s) and the source, provide a link to the Creative Commons license and
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Commons license, unless indicated otherwise in a credit line to the material. If material is not
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the copyright holder.
Chapter 6
Generalized Linear Mixed Models
for Proportions and Percentages
6.1
Response Variables as Ratios and Percentages
In this chapter, we will review generalized linear mixed models (GLMMs) whose
response can be either a proportion or a percentage. For proportion and percentage
data, we refer to data whose expected value is between 0 and 1 or between 0 and 100.
For the remainder of this book, we will refer to this type of data only in terms of
proportion, knowing that it is possible to change it to a percentage scale only when
multiplying it by 100. Proportions can be classified into two types: discrete and
continuous. Discrete proportions arise when the unit of observation consists of
N distinct entities, of which individuals have the attribute of interest “y”. N must
be a nonnegative integer and “y” must be a positive integer; here, y ≤ N. Therefore,
the observed proportion must be a discrete fraction, which can take values
0 1
N
N , N , ⋯, N . A binomial distribution is the sum of a series of m independent binary
trials (i.e., trials with only two possible outcomes: success or failure), where all trials
have the same probability of success. For binary and binomial distributions, the
target of inference is the value of the parameter such that 0 ≤ E Ny = π ≤ 1. Continuous proportions (ratios) arise when the researcher measures responses such as the
fraction of the area of a leaf infested with a fungus, the proportion of damaged cloth
in a square meter, the fraction of a contaminated area, and so on. As with the
binomial parameter π, the continuous rates (fractions) take values between 0 and
1, but, unlike the binomial, the continuous proportions do not result from a set of
Bernoulli tests. Instead, the beta distribution is most often used when the response
variable is in continuous proportions. In the following sections, we will first address
issues in modeling when we have binary and binomial data. When the response
variable is binomial, we have the option of using a linearization method (pseudolikelihood (PL)) or the Laplace or quadrature integral approximation (Stroup 2012).
© The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8_6
209
210
6.2
6
Generalized Linear Mixed Models for Proportions and Percentages
Analysis of Discrete Proportions: Binary
and Binomial Responses
A binomial distribution is the number of successes from a series of N independent
binary trials – Bernoulli trials (i.e., trials with two possible outcomes: success or
failure), where all trials have the same probability of success. In the context of a
GLMM, there are N binomial responses, each of which is the result of binary trials.
The ith response consists of two pieces of information: the number of trials ni and the
number of successes yi, as shown in the following example.
6.2.1
Completely Randomized Design (CRD): Methylation
Experiment
An agent to induce demethylation is applied to plants; this agent converts methylated
nucleotides to their unmethylated forms, thus causing epigenetic changes that
produce or induce abnormal phenotypes such as deformation or stunting (Amoah
et al. 2008). A pilot study was implemented to investigate the relationship between
the dose of the demethylating agent and the observed proportion of plants with a
normal phenotype. Seeds were treated with the demethylating agent at six different
doses, including the control. Plants were sown in trays, with each tray containing
seeds previously treated with the same dose of the demethylating agent. Each dose
was replicated 4 times: 2 with 60 plants and 2 with 100 plants. The trays were
allocated following a completely randomized design (CRD). The plants with a
normal phenotype in each tray are shown (in Table 6.1) with the number of plants
per tray (N ). The notation 59(60) indicates that 59 normal plants were found out of
60 plants under study. In the same way, the notation 14(100) indicates that 14 normal
plants were found out of 100 plants under study.
The sources of variation and degrees of freedom (DFs) for this experiment are
shown in Table 6.2.
Table 6.1 Number of normal
plants out of a total of N plants
per tray and dose of the
demethylating agent
Dose
0
59(60)
58(60)
99(100)
98(100)
Table 6.2 Sources of
variation and degrees of
freedom
Sources of variation
Dose
Error
Total
0.01
58(60)
59(60)
98(100)
99(100)
0.1
54(60)
53(60)
88(100)
87(100)
0.5
4(60)
11(60)
14(100)
15(100)
1.0
3(60)
2(60)
2(100)
1(100)
1.5
3(60)
3(60)
1(100)
3(100)
Degrees of freedom
t-1=6-1=5
t(r - 1) = 6 × (4 - 1) = 18
t × r - 1 = 6 × 4 - 1 = 23
6.2
Analysis of Discrete Proportions: Binary and Binomial Responses
211
Observed proportion
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Dose
Fig. 6.1 Effect of the demethylating agent on the proportion of normal plants
The statistical model of a completely randomized design (CRD) is
yij = μ þ τi þ εij
where yij is the number of observed normal plants in the tray j ( j = 1, 2, 3, 4) at the
dose i (i = 1, 2, ⋯, 6), μ is the overall mean, τi is the effect of dose i of the
demethylating agent, and εij are non-normal errors.
The expected value (normal plants) of a set of tests ni follows a binomial
distribution yi ~ Binomial(ni, π i), where π i is the probability of success in each
trial, with 0 ≤ π i ≤ 1, where π i = yi=ni . Thus, the probability of observing an outcome
yi can be written as
PðY i = yi jni , yi Þ =
ni
yi
π i i ð1 - π i Þni - yi ; yi = 0, 1, ⋯, ni :
y
This probability depends on the number of known tests ni, whereas the probability of success (π i) is an unknown parameter. In Fig. 6.1, we observe that the
probability of obtaining a normal plant depends on the applied dose of the
demethylating agent. Given that yi has a binomial distribution, the expected value
(the mean) is the product of the number of trials and the probability of success in
each trial, that is, E(Yi) = niπ i. Since the number of trials is fixed (once the data have
been obtained), modeling the probability of success is equivalent to modeling the
expected value as well as the variance since it is also a function of the number of
trials and the probability of success. So, the expected value and variance of yi are
212
6
Generalized Linear Mixed Models for Proportions and Percentages
E ðyi Þ = μi = ni π i ; Varðyi Þ = ni π i ð1 - π i Þ:
This variance is small if the value π i is close to 0 or 1, and this increases to its
maximum when π i = 0.5. This can be seen in Fig. 6.1, where proportions close
to 0 or 1 show less variance than do proportions between 0.1 and 0.2 for a
demethylating agent dose of 0.5. This variance can also be written in terms of the
expected value as:
Varðyi Þ =
μi
ðn - μi Þ:
ni i
In this CRD, the fixed number of treatments t (doses) were randomly assigned to
r experimental units (trays). The linear predictor describing the structure of the mean
of this GLMM is
ηi = η þ τ i
where ηi denotes the ith linear predictor, η is the intercept, and τi is the fixed effect
due to treatments i (i = 1, 2, ⋯, t) with t treatments and ri replicates in each
treatment.
The components that define this GLMM are shown below:
Distribution: yi~Binomial(Nij, π i)
Linear predictor: ηi = η + τi
Link function: logitðπ i Þ = logit
πi
1 - πi
= ηi
where ηi is the linear predictor that relates the effect of dose i (i = 1, 2, ⋯, 6) to
probability π i. The model uses the linear predictor (ηi) to estimate the means (π i = μi)
of the observations for each treatment.
The following GLIMMIX program fits a CRD with a binomial response:
proc glimmix nobound method=Laplace;
class Dose Rep;
model y/N= dose/link=logit;
lsmeans dose/lines ilink;
run;
In this example, the distribution of the dataset was not specified to GLIMMIX in
the model specification because by using the expression “y/N,” proc GLIMMIX
automatically infers that this dataset has a binomial distribution. It is also important
to note that variable dose and repetition were declared as class variables in the
“class” command, which Statistical Analysis Software (SAS) interprets as explanatory variables that are nonnumerical factors. However, the variable declared “Rep” is
not used in the model specification.
6.2
Analysis of Discrete Proportions: Binary and Binomial Responses
213
Table 6.3 Results of the analysis of variance
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
Dose
5
15
(c) Dose least squares (LS) means
Dose Estimate Standard error DF t-value
0
3.9580 0.4122
15
9.60
0.01
3.9580 0.4122
15
9.60
0.1
2.0049 0.1728
15
11.60
0.5
-1.8360 0.1623
15 -11.31
1
-3.6633 0.3580
15 -10.23
1.5
-3.4337 0.3212
15 -10.69
83.46
11.95
0.50
F-value
132.53
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
0.9813
0.9813
0.8813
0.1375
0.02501
0.03126
Pr > F
<0.0001
Standard error mean
0.007581
0.007581
0.01808
0.01925
0.008729
0.009728
Part of the results is shown in Table 6.3. Pearson’s chi-squared statistic value
divided by the degrees of freedom in part (a) (Pearson′s chi - square/DF = 0.5)
indicates that there is no evidence of extra-dispersion in the dataset. The analysis of
variance (ANOVA) tabulated in part (b) in Table 6.3, with the type III tests of fixed
effects, indicates that there is a highly significant difference (P = 0.0001) in the
average proportion of normal plants with respect to the dose applied to the seeds.
The output when using the “lsmeans” command in conjunction with the “ilink”
option is in the “Mean” column (part (c) in Table 6.3). These values are the values of
π i′s, i.e., the estimated probabilities π^0 = 0:9813 and π^0:01 = 0:9813 of normal plants
for the treatments whose doses are 0 and 0.01, respectively. For treatments with
doses of 0.1 and 0.5, the observed probabilities of normal plants are π^0:1 = 0:8813
and π^0:5 = 0:1375, respectively, whereas for the 1 and 1.5 doses, the observed
probabilities of normal plants decrease dramatically with π^1 = 0:02501 and
π^1:5 = 0:03126, respectively.
Figure 6.2 shows the mean comparisons (least significance difference (LSD)) of
the estimated probabilities according to the dose applied to the seeds in trays. In this
figure, we can observe that in the treatments with dose = 0 (control) and dose = 0.01,
the observed proportions of normal plants are not statistically different from each
other, but they do differ with the other applied doses. At a dose of 0.1, the observed
proportion of normal plants was 88.13%, and this was statistically different from all
the doses used. Finally, doses at 0.5, 1, and 1.5 of the demethylating agent in the
observed proportion of normal plants decreased drastically to 13.75%, 2.501%, and
3.12%, respectively. The doses of 1 and 1.5 produced statistically equal proportions
of normal plants.
214
6
Generalized Linear Mixed Models for Proportions and Percentages
Mean
1.1
Average roportion
0.825
0.55
0.275
0.
C
0.01
0.1
0.5
1
1.5
Dose
Fig. 6.2 Comparison of the estimated probabilities per dose of the demethylating agent
If the researcher wishes to model how dose levels of the demethylating agent
affect normal plant proportions, then the dose must be declared as a continuous
variable. The following SAS syntax with proc GLIMMIX runs a binomial
regression:
proc glimmix data=crd_bin method=Laplace plots=all;
class Rep;
model y/N= dose/solution;
random rep;
run;quit.
Most of the commands and options have already been discussed throughout this
book; the “model y/N” command indicates that the response variable is in a ratio.
Therefore, this dataset is modeled with a binomial distribution, which is affected by
the different number of individuals in each repetition. proc GLIMMIX interprets the
distribution of the data as binomial, whereas the “solution” option requests the
parameter estimates of the model (intercept and slope).
The components that define this GLMM are shown below:
Distribution: yi~Binomial(Nij, π i)
Linear predictor: ηi = η + β dosei
Link function: logitðπ i Þ = logit
πi
1 - πi
Thus, the model can be written as
= ηi
6.2
Analysis of Discrete Proportions: Binary and Binomial Responses
215
Table 6.4 Regression analysis results
(a) Fit statistics
-2 Log likelihood
Akaike information criterion (AIC) (smaller is better)
AICC (smaller is better)
Bayesian information criterion (BIC) (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
Dose
1
19
(c) Solutions for fixed effects
Effect
Estimate
Standard error
Intercept
2.7927
0.1302
Dose
-7.6232
0.3494
ηi = log
231.58
235.58
236.15
237.93
239.93
236.20
2317.12
96.55
F-value
475.97
DF
3
19
t-value
21.46
-21.82
Pr > F
<0.0001
Pr > |t|
0.0002
<0.0001
μi
ni π i
πi
= log
= log
= logitðπ i Þ = η þ βdosei
ni - μi
ni - ni π i
1 - πi
and the logit function can be written in terms of the probability of success, π i, as
πi =
1
1 þ expð- ηi Þ
Part of the SAS output of the GLIMMIX syntax is shown below. The goodnessof-fit statistics, type III tests of fixed effects, and parameter estimates are shown in
Table 6.4. The analysis of variance indicates that the demethylating agent has a
highly significant effect on the observed proportion of normal plants (P < 0.0001)
(part (b)). The maximum likelihood estimates for the intercept and slope are
η = 2.7927 and β = - 7.6232, respectively.
Figure 6.3 shows that as the value of the linear predictor increases (ηi), the value
of the residuals rapidly decreases. We can also see that the residuals plotted against
the quantiles clearly do not follow a normal distribution because this model is not a
linear function of the explanatory variable “dose.”
Figure 6.4 shows that the proportions studied and fitted are not so far apart, and,
as such, the binomial model is suitable for this dataset. The estimated linear predictor
of this model is as follows:
^ηi = ^η þ β^ × dosei = 2:7927 - 7:6232 × dosei :
216
6
Generalized Linear Mixed Models for Proportions and Percentages
Residuals
80
60
200
Percent
Residual
300
100
40
20
0
0
-7.5
-5.0
-2.5
0.0
Linear Predictor
2.5
-240 -120
0 120 240 360
Residual
21
22
300
300
Residual
Residual
200
100
0
200
24
100
23
-100
0
-2
-1
0
Quantile
1
2
Fig. 6.3 A graph of residuals versus the linear predictor, quantiles
1.1
Probability ( i)
0.825
0.55
0.275
0.
0.
0.4
0.8
1.2
1.6
Dose
Fig. 6.4 Observed and estimated proportion
The logit of the probability of success is a linear function of the explanatory
variables, so the model can be written in terms of the probability of success
(observing normal plants) as
Factorial Design in a Randomized Complete Block Design (RCBD). . .
6.3
πi =
217
1
1 þ exp ð- ηi Þ
Given the parameter estimates, we can predict the success probability of observing a normal plant, and given a certain concentration of the demethylating agent, this
estimated probability (using the estimated linear predictor) can be seen plotted in
Fig. 6.4.
π^i =
6.3
1
1
=
1 þ expð^ηi Þ 1 þ expð - 2:7927þ7:6232 × dosei Þ
Factorial Design in a Randomized Complete Block
Design (RCBD) with Binomial Data: Toxic Effect
of Different Treatments on Two Species of Fleas
A group of researchers wishes to study the toxic effect of certain treatments (Trts) on
two flea species (SP) (Daphnia magna and Ceriodaphnia dubia). To compare the
toxicity effect of treatments on both flea species, a randomized complete block
design (RCBD bioassay) was implemented with three replicates per treatment,
with each replicate consisting of 10 fleas (Appendix: Fleas). The linear predictor
describing this experiment is described below:
ηijkl = η þ αi þ βj þ ðαβÞij þ bioassayk þ repðbioassayÞlðkÞ
where η is the intercept, αi is the fixed effect due to species i, βj is the fixed effect
of treatment j, (αβ)ij is the fixed effects interaction between the flea species and
treatment, bioassayk is the random effect due to bioassay k assuming
bioassayk N 0, σ 2bioassay , and rep(bioassay)l(k) is the random effect due to repetition bioassay assuming repðbioassayÞlðkÞ N 0, σ 2repðbioassayÞ .
The remaining components of this GLMM with a binomial response (Nijk, π ijk) are
described below:
Distribution: yijkl j bioassayk, rep(bioassay)l(k)~Binomial(Nijk, π ijk)
bioassayk N 0, σ 2bioassay , repðbioassayÞlðkÞ N 0, σ 2repðbioassayÞ , where Nijkl is
the number of dead fleas, observed in species i in replicate l in bioassay k under
treatment j,
Link function: logit π ijk = log
π ijk
1 - π ijk
= ηijk .
The following SAS syntax allows us to fit the GLMM with a binomial response.
218
Table 6.5 Results of the
analysis of variance
6
Generalized Linear Mixed Models for Proportions and Percentages
(a) Fit statistics
-2 Log likelihood
145.33
AIC (smaller is better)
173.33
AICC (smaller is better)
177.85
BIC (smaller is better)
160.71
CAIC (smaller is better)
174.71
HQIC (smaller is better)
147.97
(b) Fit statistics for conditional distribution
-2 Log L (Sobrevi | r. effects)
145.33
Pearson’s chi-square
10.72
Pearson’s chi-square/DF
0.10
(c) Covariance parameter estimates
Cov Parm
Estimate
Standard error
Bioen
-0.1051
.
Bioen*SP (Rep)
-0.1192
.
(d) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
SP
1
14
0.02
0.8829
Trt
5
80
15.08
<0.0001
SP*trt
5
80
4.66
0.0009
proc glimmix data=pulgas nobound method=laplace;
class Bioen SP Trt Rep ;
Model Sobrevi/n = SP|Trat/dist=binomial;
random Bioen sp*bioen(rep);
lsmeans SP|Trt/lines ilink;
run;
Part of the results is listed in Table 6.5. The fit statistics in part (a) and the
conditional statistics in part (b) are useful for model comparison, whereas the
variance component estimates are shown in part (c). The value of the statistic
Pearson’s chi - square/DF = 0.10 indicates that the binomial model gives a good
fit to the dataset. The variance component estimates for bioassays and replication
nested in bioassays are σ^2bioassay = - 0:1051 and σ^2repðbioassayÞ = - 0:1192, respectively. The type III tests of fixed effects (part (d)) show the significance tests of the
fixed effects in the model. The treatment effect and the interaction between the flea
species (SP) and treatment are clearly significant with P < 0.0001 and P = 0.0009,
respectively.
Since survival was statistically similar in both flea species, we will focus on the
factors that were significant. Part (a) in Table 6.6 shows the means and standard
errors of treatments on the model scale (“Estimate” column) and on the data scale
(“Mean” column), obtained with “lsmeans” and the “ilink” option as well as the
mean comparisons, which are on the model scale (part (b)).
6.4
A Split-Plot Design in an RCBD with a Normal Response
219
Table 6.6 Means and standard errors on the model scale and on the data scale
(a) Trt least squares means
Trt Estimate Standard error DF t-value Pr > |t|
Mean
T1
8.1179 4.3180
80
1.88
0.0637 0.9997
T2
4.3564 3.0554
80
1.43
0.1578 0.9873
T3
1.0081 0.1924
80
5.24 <0.0001 0.7326
T4 -1.0509 0.1712
80
-6.14 <0.0001 0.2591
T5 -4.7187 3.0570
80
-1.54
0.1266 0.008848
T6 -8.1182 4.3184
80
-1.88
0.0638 0.000298
(b) Conservative T grouping of Trt least squares means (α=0.05)
LS means with the same letter are not significantly different
Trt
Estimate
T1
8.1179
A
T2
4.3564
B
A
T3
1.0081
B
A
T4
-1.0509
B
D
T5
-4.7187
D
T6
-8.1182
D
Standard error mean
0.001287
0.03820
0.03768
0.03286
0.02681
0.001286
C
C
C
The LINES display does not reflect all significant comparisons. The following additional pairs are
significantly different: (T3,T4)
Based on the fixed effects tests, the flea species × treatment interaction is
significant. The means on the model scale are listed under the “Estimate” column,
followed by their standard errors, “Standard error” (Table 6.7). The output of the
“ilink” option in “lsmeans” applies the inverse function of the link function to the
estimates on the model scale to obtain the estimates on the data scale. The probabilities, on the data scale, are given under the “Mean” column with their respective
standard errors and correspond to the probability of insect (flea) survival.
Figure 6.5 shows that the survival of both species is different in treatments 2–5;
the Daphnia species showed more resistance in treatments 2 and 3, whereas the
Ceriodaphnia species showed greater resistance in treatments 4 and 5. On the other
hand, in treatments 1 and 6, survival was similar in both species.
6.4
A Split-Plot Design in an RCBD with a Normal
Response
A split plot is the most common treatment structure design in agricultural and agroindustrial research areas. These experiments generally involve two or more factors
under study. Typically, large or primary experimental units, commonly known as the
whole plot, are grouped into blocks. The levels of the first factor are randomly
assigned to the whole plots. Then, each whole plot is divided into smaller units,
known as split or secondary plots. The levels of the second factor are randomly
assigned to the subplots within each whole plot.
SP*treatment least squares means
SP
Treatment
Daphnia
T1
Daphnia
T2
Daphnia
T3
Daphnia
T4
Daphnia
T5
Daphnia
T6
Ceriodaphnia
T1
Ceriodaphnia
T2
Ceriodaphnia
T3
Ceriodaphnia
T4
Ceriodaphnia
T5
Ceriodaphnia
T6
Estimate
8.1179
8.1180
1.9717
-1.2537
-8.1186
-8.1182
8.1178
0.5947
0.04446
-0.8480
-1.3188
-8.1182
Standard error
6.1065
6.1068
0.3218
0.2536
6.1085
6.1073
6.1064
0.2202
0.2109
0.2301
0.2583
6.1071
DF
80
80
80
80
80
80
80
80
80
80
80
80
t-value
1.33
1.33
6.13
-4.94
-1.33
-1.33
1.33
2.70
0.21
-3.69
-5.11
-1.33
Pr > |t|
0.1875
0.1875
<0.0001
<0.0001
0.1876
0.1875
0.1875
0.0084
0.8336
0.0004
<0.0001
0.1875
Table 6.7 Means and standard errors on the model scale and on the data scale of the interaction between both factors
Mean
0.9997
0.9997
0.8778
0.2221
0.0002
0.0002
0.9997
0.6444
0.5111
0.2999
0.2110
0.0002
Standard error mean
0.001820
0.001820
0.03452
0.04381
0.001819
0.001819
0.001820
0.05046
0.05269
0.04830
0.04301
0.001819
220
6
Generalized Linear Mixed Models for Proportions and Percentages
6.4
A Split-Plot Design in an RCBD with a Normal Response
221
Survival probability
1.1
0.825
0.55
0.275
0.
Trt1
Trt2
Trt3
Trt4
Trt5
Trt6
Treatment
Dapnhia
Ceriodaphnia
Fig. 6.5 The average survival rate of both species
The model equation for the analysis of variance assuming normality in the
response is
yijk = η þ αi þ r k þ ðraÞik þ βj þ ðαβÞij þ eijk
i = 1, 2, ⋯, a; j = 1, 2, ⋯, b; k = 1, 2, ⋯, r
where yijk is the observed response variable in the kth block at the ith level of factor A
and at the jth level of factor B, α and β refer to the fixed treatment effects due to
factors A and B, respectively, r is the random effect due to the blocks, (ra)ik is the
random error term due to the whole plot that is an interaction between the blocks and
factor A, and eijk is the random residual effect. Normally, the errors and other random
terms are also assumed to be normal; however, when the response variable is not
normally distributed, this way of specifying the model is not the most appropriate.
Thus, under the assumption that the response variable is normal, this way of
specifying the model is valid.
6.4.1
An RCBD Split Plot with Binomial Data: Carrot Fly
Larval Infestation of Carrots
Data were obtained from an experiment that was designed to compare a number of
carrot genotypes with respect to their resistance to infestation by carrot fly larvae.
The data involved 16 genotypes that were compared at 2 pest levels to be controlled.
The experiment was conducted in three randomized blocks. Each block consisted of
222
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.8 The notation 44/53 denotes that 44 carrots were infected ( y) out of a sample size of
53 studied (N )
Genotype
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
G11
G12
G13
G14
G15
G16
Treatment (level of infestation)
1
Block1
Block2
Block3
44/53
42/48
27/51
24/48
35/42
45/52
8/49
16/49
16/50
4/51
5/42
12/46
11/52
13/51
15/44
15/50
5/49
7/50
18/52
13/47
7/47
5/47
15/49
8/50
11/52
6/45
5/51
0/51
10/39
14/48
6/52
4/46
10/37
0/52
4/55
1/40
14/45
18/43
4/40
3/52
12/53
4/55
11/52
6/54
5/49
4/53
1/40
4/52
2
Block1
16/60
13/44
4/52
15/52
4/51
1/51
2/52
6/56
3/54
3/50
1/52
1/50
4/51
3/52
2/50
4/56
Block2
9/52
20/48
6/51
10/56
6/43
8/49
4/52
4/50
8/51
0/50
7/38
3/50
7/46
7/48
4/46
1/44
Block3
26/54
16/53
12/43
6/48
9/46
3/54
6/52
6/42
3/53
10/51
4/48
1/45
7/45
12/49
14/53
3/42
Table 6.9 Sources of variation and degrees of freedom
Sources of variation
Blocks
Factor A (infestation)
Errora (A*blocks)
Factor B (genotypes)
Infestation*genotype (A*B)
Errorb
Total
Degrees of freedom
r-1=3-1=2
a-1=2-1=1
(r - 1)(a - 1) = 2
b - 1 = 16 - 1 = 15
(a - 1)(b - 1) = 15
a(r - 1)(b - 1) = 2 × 2 × 15 = 60
r × a × b - 1 = 3 × 2 × 16 - 1 = 95
32 plots, 1 for each combination of genotype and pest infestation level. At the end of
the experiment, about 50 carrots were taken from each plot and assessed for
infestation by carrot fly larvae. The data obtained are shown in Table 6.8.
Table 6.9 shows the analysis of variance summarizing the sources of variation
and degrees of freedom.
Rewriting in terms of the linear predictor
ηijk = η þ αi þ r k þ ðraÞik þ βj þ ðαβÞij
Since the observations were taken at the subplot level, conditioned on the
structural effects of the design, these observations have a variance associated with
the subplot. Therefore, α and β refer to the treatment fixed effects due to factors A
6.4
A Split-Plot Design in an RCBD with a Normal Response
Table 6.10 Results of the
analysis of variance
223
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
527.82
Pearson’s chi-square
189.09
Pearson’s chi-square/DF
1.97
(b) Covariance parameter estimates
Cov Parm
Subject
Estimate
Standard error
Intercept
Bloque
0.004272
0.02741
Trt
Bloque
0.03344
0.03545
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Genotype
15
60
28.28
<0.0001
Trt
1
2
16.24
0.0564
Genotype*Trt
15
60
4.45
<0.0001
and B, respectively; (αβ)ij refers to the interaction of the above factors; rk is the
random effect due to blocks; and blocks × whole plot (ra)ik is assumed to contribute
to the variation such that r k N 0, σ 2r and ðraÞik N 0, σ 2block × A . This model uses
the linear predictor ηijk to estimate the mean of the observations μijk.
The specification of the this GLMM is as follows:
Distribution: yijk j rk, (ra)rk~Binomial(Nijk, π ijk)
r k N 0, σ 2r ,
ðraÞrk N 0, σ 2blockA
Link function: logit(π ijk) = ηijk.
The following SAS GLIMMIX program allows the fitting of a GLMM with a
split-plot structure in a randomized complete block design with a binomial response.
proc glimmix data=spd_pp nobound method=quadrature;
class Genotype Trt Block ;
model y/N = Genotype|Trt;
random intercept trt /subject=block;
lsmeans Genotype|Trt/lines ilink;
run;
The program uses the quadrature estimation method (method=quadrature).
This estimation method produces similar results as the Laplace method. Part of the
results is provided in Table 6.10. Pearson’s chi-squared/DF value in part (a) gives an
idea of whether there is overdispersion or extra-variation in the dataset. In this case,
Pearson’s chi - square/DF = 1.97 indicates that there is overdispersion in the
dataset, so it is feasible to use either the pseudo-likelihood (PL) estimation method
or a different distribution. In addition to these results, the variance component
estimated due to blocks and blocks × genotype (the whole plot) in part (b) are
σ 2block = 0:004272 and σ ð2block × AÞ = 0:03344, respectively. The results of the fixed
224
6 Generalized Linear Mixed Models for Proportions and Percentages
effects tests (part (c)) indicate that the effect of genotype and the interaction between
genotype and treatment are significant.
The appropriate method for model evaluation depends on whether or not there is
evidence of overdispersion, so we consider this issue below. The residual variance
incorporates systematic discrepancies between the model and the observed
responses, variation between replicates (observations in independent experimental
units with the same values of the explanatory variables) and sampling variation
arising from the distribution of the data; in this case, it is the binomial distribution. If
there are no duplicate observations and the fitted model provides an adequate
description of the systematic trend, then only sampling variation contributes to the
residual variance. If this is true, then the residual deviation has an approximate
chi-squared distribution with degrees of freedom similar to the mean squared error
(MSE) (the residual).
Since there is overdispersion in the data using the binomial distribution, there are
three alternatives we can explore: (1) review the linear predictor, which involves
carefully revising the analysis of variance table; (2) add a scale parameter; or (3) use
another distribution for the dataset. Each of these three possible alternatives is
discussed below, in this order.
6.4.1.1
Linear Predictor Review (ηijk)
If the proportion of normal plants (π ijk) is being affected by the genotype within each
infestation level (trt = αi) from plot to plot within each of the blocks, then a nested
factorial effect of genotype within infestation levels (trt) could be included in the
analysis of variance. Thus, the linear predictor would be defined as
ηijk = η þ αi þ r k þ ðraÞik þ βðαÞjðiÞ
where αi, β(τ)j(i), rk, and (ra)ik are the fixed effects due to treatments, the effect of
genotypes nested within a treatment, random effects due to blocks r k N 0, σ 2r ,
and the interaction between blocks and treatment ðraÞik N 0, σ 2RA , respectively.
The following GLIMMIX syntax estimates the above linear predictor:
proc glimmix data=spd_pp method=laplace;
class Genotype Trt Block ;
model y/N = Trt genotype(trt);
random trt/subject=block;
lsmeans genotype(trt)/lines ilink slice=trt slicediff=trt;
run;
The only difference between this proc GLIMMIX and the previous one is that in
this program, we have included the nested effect of genotypes within treatment,
genotype (trt), and removed only the fixed effects of genotypes. Part of the results is
shown in Table 6.11. The value of Pearson’s chi-squared/DF statistic (part (a)) as
6.4
A Split-Plot Design in an RCBD with a Normal Response
Table 6.11 Results of the
analysis of variance, under a
new linear predictor
225
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
527.82
Pearson’s chi-square
189.07
Pearson’s chi-square/DF
1.97
(b) Covariance parameter estimates
Cov Parm
Subject
Estimate
Standard error
Intercept
Bloque
0.004265
0.02740
Trt
Bloque
0.03343
0.03544
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Trt
1
2
16.20
0.0565
Genotype (Trt)
30
60
15.83
<0.0001
well as the fit statistics did not decrease when modifying the linear predictor.
However, the F-values calculated for treatments and genotypes within treatments
(part (c)) are smaller than those obtained in the split-plot design.
Since the overdispersion is still present (Pearson’s chi - square/DF = 1.97),
another alternative is to add a scaling parameter to the model. This alternative is
presented below.
6.4.1.2
Scale Parameter
If the residual deviation is larger than expected when compared to critical values of
the appropriate chi-squared distribution, and if this cannot be corrected by redefining
the linear predictor of the model, then there is more variation present than can be
accounted for by the distributional likelihood assumption. In this case, we say that
the data show overdispersion. The simplest way to deal with overdispersion is to
extend the model for scaling the variance function. Adding the scale parameter
replaces Var(yij) = π ij(1 - π ij) with Var(yij) = ϕπ ij(1 - π ij). The rationale for this
approach is discussed by Collett (2002). The parameter ϕ is a scale factor, called the
dispersion parameter, which is used to summarize the degree of overdispersion
present in the observations. Clearly, ϕ = 1 corresponds to the original distribution
model. This parameter can be estimated in several different ways. The logarithm of
the likelihood of the binomial distribution is given by
log
N
yij
þ yij log
π ij
1 - π ij
þ N log 1 - π ij
In the logarithm of the likelihood, the term “yij log
π ij
1 - π ij
” is very important; any
quantity that multiplies yij is known as the natural or canonical parameter, and this
parameter is always a function of the mean. For the binomial distribution, the mean
226
6 Generalized Linear Mixed Models for Proportions and Percentages
Nijπ ij and the natural parameter is log
π ij
1 - π ij
, and, in categorical data, it is known as
“log odds.” The generalized estimating equation (GEE) method provides a valid
analysis for marginal means, since under a binomial distribution, in the quasilikelihood, the variance of the distribution is given by ϕπ ij(1 - π ij). This is achieved
by adding the “random _residual_” command in the following SAS syntax.
The following GLIMMIX commands are used to invoke the scale parameter but
using the first predictor proposed for these data.
proc glimmix data=spd_pp nobound;
class GenotypeTrtBlock ;
model y/N = Trt|genotype;
random intercept trt/subject=block;
random _residual__;
lsmeans Trt|genotype/lines ilink ;
run;
In this syntax, we still keep the binomial distribution (y/N is equivalent to telling
GLIMMIX in SAS that it is a binomial response) but will add the “random
_residual_” command. In this case, we cannot obtain the maximum likelihood
estimators because we cannot implement the Laplace method (“method = laplace”)
or adaptive quadrature (“method = quad”) approximation method, so the estimation
is performed through the pseudo-likelihood (PL) method. This causes the scale
parameter to be estimated, and, consequently, it is used in the adjustment of all
standard errors and statistical tests. Proc GLIMMIX uses the generalized statistics of
^
McCullagh and Nelder (1989), i.e., χ 2/df as the estimator of the scale parameter (ϕÞ.
All standard errors from the analysis under a binomial distribution are multiplied by
^ and all F-tests are divided by ϕ
^ to account for overdispersion. Part of the output
ϕ,
is shown below.
The value of Pearson’s statistic in part (a) indicates that overdispersion has not
been eliminated. Chi - square/DF = 3.13, on the contrary, indicates that this value
has increased. This result indicates that adding a scale parameter to the model does
not decrease the extra-variation present in the dataset, since the binomial assumption
forces a relationship between the mean and variance of the data that might not
contain the data being analyzed. On the other hand, the estimated scale parameter is
^ = 3:1263 (part (b)). Pearson’s residual analysis showed that its variance is 3.6257,
ϕ
which is considerably larger than 1, implying a large overdispersion. In addition, the
results of the fixed effects tests (part (c)) vary from those above (Table 6.12).
Therefore, the third option based on assuming an alternative distribution (beta
distribution) on the response variable is discussed below.
6.4
A Split-Plot Design in an RCBD with a Normal Response
Table 6.12 Results of the
analysis of variance, adding a
scale parameter to the model
(a) Fit statistics
-2 Res log pseudo-likelihood
Generalized chi-square
Gener. chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Subject
Intercept
Bloque
Trt
Bloque
Residual variance component (VC)
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
Trt
1
2
Genotype
15
60
Genotype*Trt
15
60
6.4.1.3
227
182.52
200.09
3.13
Estimate
0.005416
0.03202
3.1263
F-value
10.20
9.04
1.42
Standard
error
0.04750
0.06338
0.5719
Pr > F
0.0856
<0.0001
0.1674
Alternative Distribution
Another approach to control the overdispersion would be to use a different distribution in the interval [0, 1], such as the beta distribution, to model the data.
Generally, this distribution yields good results when all experiments have the
same number of observations (successes and failures), i.e., when Nijk = N. When
Nijk varies a little, even in many cases, the beta distribution yields acceptable results.
It is important to mention that the proportions come from binomial counts, and,
y
therefore, we now define the response variable as pijk = Nijkijk so that it can be modeled
as the beta distribution. The components of the beta response model are listed below:
Distribution: pijk j rk, (ra)rk~Beta(π ijk, ϕ) with ϕ as the scale parameter
r k N 0, σ 2r , ðraÞrk N 0, σ 2RA
Linear predictor: ηijk = η + αi + rk + (αr)ik + βj + (αβ)ij
Link function: logit π ijk = logit
π ijk
1 - π ijk
= ηijk
y
As mentioned before, we now use the response variable pijk = Nijkijk . This new
response variable pijk is not the same as the one used in the binomial distribution. The
following SAS commands fit a GLMM in a split-plot randomized complete block
design with a beta response. It is important to mention that before implementing this
y
model in SAS GLIMMIX, the variable p = pijk = Nijkijk was defined.
proc glimmix data=spd_pp nobound method=laplace;
class GenotypeTrtBlock ;
model p = Genotype|Trt/dist=beta;
random intercept trt/subject=block;
lsmeans Genotype|Trt/lines ilink;
run;
228
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.13 Fit statistics
assuming binomial and beta
distributions
(a) Fit statistics
Distribution
Binomial
-2 Log likelihood
541.85
AIC (smaller is better)
609.85
AICC (smaller is better)
648.87
BIC (smaller is better)
579.20
CAIC (smaller is better)
613.20
HQIC (smaller is better)
548.24
(b) Fit statistics for conditional distribution
Distribution
Binomial
-2 Log L (y | r. effects)
527.82
Pearson’s chi-square
189.09
Pearson’s chi-square/DF
1.97
Beta
-246.49
-176.49
-132.28
-208.04
-173.04
-239.91
Beta
-254.68
93.95
1.01
Table 6.14 Results of the analysis of variance, assuming binomial and beta distributions
(a) Covariance parameter estimates
Binomial
Cov Parm
Subject
Estimate
Intercept
Bloque
0.004272
Trt
Bloque
0.03344
^
Scale ϕ
Standard error
0.02741
0.03545
.
Beta
Estimate
-0.00524
0.02175
25.7070
Standard error
.
0.1475
(b) Type III tests of fixed effects
Effect
Trt
Genotype
Genotype*Trt
Num DF
1
15
15
Den DF
4
60
60
Binomial
F-value
16.24
28.28
4.45
Pr > F
0.0564
<0.0001
<0.0001
Beta
F-value
9.98
13.25
2.23
Pr > F
0.0342
<0.0001
0.0146
Some of the SAS GLIMMIX output is listed below. Based on the fit statistics
under the binomial (first alternative) and beta distributions (Table 6.13), clearly the
values of the statistics related to the degree of overdispersion are lower in the beta
distribution than in the binomial distribution, indicating that the beta distribution
provides a better fit (part (a)). Looking at the fit statistics for the conditional model in
part (b), the values of the three fit statistics in the binomial model are higher than the
values in the beta model. The value of Pearson’s chi - square/DF under the beta
distribution is 1.01. This value indicates that the overdispersion has been virtually
eliminated from the data and that therefore the beta distribution is a better candidate
model for this dataset.
Adding the scale parameter (ϕ) to the model, the variance components and
standard errors in Table 6.14 cause (part (a)) variation for each of the results and,
therefore, the F- and t-tests are affected (part (b)). The estimated value of the scale
6.4
A Split-Plot Design in an RCBD with a Normal Response
229
Table 6.15 Estimated means and standard errors on the model scale and the data scale
(a) Trt least squares means
Trt
Estimate Standard error
Trt1 -1.2362 0.01768
Trt2 -1.9327 0.01768
(b) Genotype least squares means
Standard
Genotype Estimate error
G1
0.1524 0
G10
-1.4143 0
G11
-1.8698 0
G12
-2.8971 0.03885
G13
-1.4336 0
G14
-1.8761 0.1304
G15
-1.8618 0
G16
-2.6686 0
G2
0.2225 0
G3
-1.3329 0
G4
-1.5897 0
G5
-1.3696 0
G6
-2.0173 0
G7
-1.7001 0.1356
G8
-1.7161 0
G9
-1.9796 0
DF
2
2
t-value
-69.94
-109.34
DF
57
57
57
57
57
57
57
57
57
57
57
57
57
57
57
57
t-value
Infty
-Infty
-Infty
-74.58
-Infty
-14.39
-Infty
-Infty
Infty
-Infty
-Infty
-Infty
-Infty
-12.53
-Infty
-Infty
Pr > |t|
0.0002
<0.0001
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
0.2251
0.1264
Mean
0.5380
0.1956
0.1336
0.05230
0.1925
0.1328
0.1345
0.06485
0.5554
0.2087
0.1694
0.2027
0.1174
0.1545
0.1524
0.1214
Standard error mean
0.003083
0.001952
Standard error
mean
0
0
0
0.001925
0
0.01502
0
0
0
0
0
0
0
0.01771
0
0
^ = 25:7018. The variance components based on the binomial model
parameter is ϕ
and beta are listed below.
The treatment means (part (a)) and genotypes (part (b)) are presented in
Table 6.15. The estimates on the model scale are listed under the column “Estimate”
with their respective standard errors “Standard error,” and the values on the data
scale are listed under the column “MEAN” with their respective standard errors
“Standard error mean.” In the table of least squares means for the effect of genotypes, inconsistencies are observed in the values of t and in the standard error values
of the means, so other estimation alternatives should be sought.
In large samples, both binomial and normal distributions are quite similar.
Logically, the latter two analyses, binomial and beta, are attractive because of their
consistency with the nature of the data. Because of the inconsistencies in the
estimates of the mean for genotypes (tvalue = Infty and standard error of the
mean), a robust method of estimation could be used; in this case, this is the normal
distribution.
Assuming that pijk has a normal distribution with a mean μijk and constant
variance σ 2, the components of this model are as follows:
230
6
Table 6.16 Results of the
analysis of variance, assuming
a normal distribution
Generalized Linear Mixed Models for Proportions and Percentages
(a) Fit statistics
-2 Res log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Generalized chi-square
Gener. chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Estimate
Bloque
0.000123
Trt*bloque
0.000329
Residual
0.009442
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
Genotype
15
60
Trt
1
2
Genotype*Trt
15
60
-79.38
-73.38
-72.98
-76.08
-73.08
-78.81
0.60
0.01
Standard error
0.000742
0.000925
0.001724
F-value
12.59
20.46
2.93
Pr > F
<0.0001
0.0456
0.0016
Distribution: pctijk j rk, (ra)ik~Normal(μijk, σ 2)
r k N 0, σ 2r , ðraÞik N 0, σ 2RA
Linear predictor: ηijk = η + αi + rk + (αr)ik + βj + (αβ)ij
Link function: ηijk = μijk; identity
y
Similarly, in this example, the response variable used was pctijk = Nijkijk . This new
response variable pctijk is not the same as the response variable used in the binomial
distribution. The following SAS GLIMMIX commands adjust a linear mixed model
(LMM) under a split plot in a randomized complete block design with a normal
response.
proc glimmix data=spd_pct nobound;
class Genotype Trt Block ;
model pct = Genotype|Trt;
random block block*trt;
lsmeans Genotype|Trt/lines;
run;
Part of the results is shown below. The values of fit statistics in part (a) of
Table 6.16 for the model are clearly lower than those estimated in the previous
options. This indicates that the normal distribution is reasonable, even though the
response is a proportion. The estimated variance components, tabulated in
part (b) due to blocks, blocks x treatment, and the mean squared error (MSE)
(Residual = Gener. chi-square/DF) are σ^2block = 0:000123, σ^2block × trt = 0:00039, and
σ^2 = MSE = 0:009442 ffi 0:01, respectively.
6.4
A Split-Plot Design in an RCBD with a Normal Response
231
Table 6.17 Means and standard errors for genotypes and treatments
(a) Genotype least squares means
Standard
Genotype Estimate error
G1
0.5260
0.04086
G10
0.1340
0.04086
G11
0.1522
0.04086
G12
0.03332 0.04086
G13
0.2026
0.04086
G14
0.1342
0.04086
G15
0.1360
0.04086
G16
0.05625 0.04086
G2
0.5355
0.04086
G3
0.2139
0.04086
G4
0.1751
0.04086
G5
0.2035
0.04086
G6
0.1301
0.04086
G7
0.1671
0.04086
G8
0.1504
0.04086
G9
0.1187
0.04086
(b) Trt least squares means
Trt
Estimate Standard error
Trt1 0.2478
0.01863
Trt2 0.1358
0.01863
DF
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
DF
2
2
tvalue
12.87
3.28
3.73
0.82
4.96
3.28
3.33
1.38
13.11
5.24
4.28
4.98
3.18
4.09
3.68
2.90
t-value
13.30
7.29
Pr > |t|
<0.0001
0.0017
0.0004
0.4179
<0.0001
0.0017
0.0015
0.1737
<0.0001
<0.0001
<0.0001
<0.0001
0.0023
0.0001
0.0005
0.0051
Pr > |t|
0.0056
0.0183
Mean
0.5260
0.1340
0.1522
0.0333
0.2026
0.1342
0.1360
0.0562
0.5355
0.2139
0.1751
0.2035
0.1301
0.1671
0.1504
0.1187
Mean
0.2478
0.1358
Standard error
mean
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
0.04086
Standard error mean
0.01863
0.01863
The F-statistics for the fixed effects of genotype, treatments, and the interaction
between both factors provide significant statistical evidence on the proportion of
infested carrots in each of the genotypes (part (c)). Overall, the least squares means
for genotypes and treatments are reported in Table 6.17 in parts (a) and (b). The
genotypes showing the highest fraction of infested carrots were 1, 2, 3, 5, and
13, whereas genotypes 12 and 16 showed the lowest percentage of infested carrots.
Now, for treatments, the highest proportion of infested carrots was observed in
treatment 1 with 24.78%, whereas in treatment 2, it was 13.58%.
Based on the fixed effects tests, the interaction effect of genotype x treatment on
the proportion of infested carrots was statistically different. Genotypes 9 and
16 showed higher susceptibility in treatment 1 followed by treatment 2, whereas
genotypes 5, 11, 13, and 15 showed the same proportions of infested carrots in both
treatments (Fig. 6.6). On the other hand, genotypes that showed higher resistance to
infestation levels were genotypes 1, 2, and 6 followed by genotypes 3, 4, 7, 8,
10, and 12.
232
6
Generalized Linear Mixed Models for Proportions and Percentages
Proportion of infested carrots
1.
0.75
0.5
0.25
0.
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
G11
G12
G13
G14
G15
G16
Interaction
Fig. 6.6 The average proportion of infested carrots in genotypes as a function of treatment
6.5
A Split-Split Plot in an RCBD:- In Vitro Germination
of Seeds
The growth of a plant in a tissue culture can be explained by various combined
effects of A, B, and C factors. For this, the availability and efficient use of chemical
resources (factors) is of great relevance when availability is scarce or too expensive.
In light of this, the combination of three reagents (A, B, and C), reagent A at three
levels and reagents B and C at two levels, were tested on the in vitro germination of
orchid seeds. The combination of the levels of each of the factors is schematized
below.
Block 1
A3
B1
C2
C2
C1
C1
B2
C2
C1
Block 2
A2
B1
C1
C1
C2
C2
B2
C1
C2
C2
C1
A1
B1
C2
C1
C1
C2
A1
B1
C1
C2
C2
C1
B2
C2
C1
C1
C2
B2
C1
C2
C2
C1
A2
B2
C2
C1
C2
C1
B1
C2
C1
C2
C1
C1
C2
A3
B2
C1
C2
C1
C2
B1
C1
C2
C1
C2
In each of the factor combinations, N orchid seeds were placed to germinate for a
period of time. Let yijk be the number of seeds germinated at the ith level of factor A,
at the jth level of factor B, and at the kth level of factor C. Since the observations are
made at the sub-subplot level, conditional on the structural effects of the design,
these observations have a variance associated with the subplot. Therefore, the
statistical model for this experiment is given below:
6.5
A Split-Split Plot in an RCBD:- In Vitro Germination of Seeds
Table 6.18 Number of seeds
that germinated (yijkl) in each
of the factor combinations
Block
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
A
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
B
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
233
C
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
Y
15
10
17
19
26
21
14
12
10
12
30
32
37
30
32
37
18
18
23
21
24
27
37
37
N
73
86
69
32
125
62
81
21
92
108
44
33
91
42
98
44
52
73
108
55
106
92
64
97
Distribution: yijkl j rl, (ra)il, (rαβ)ijl~Binomial(Nijk, π ijk)
r l N 0, σ 2r , ðraÞrk N 0, σ 2RA , ðrαβÞijl N 0, σ 2rab
Linear predictor:
ηijk = η + αi + rl + (rα)il + βj + (αβ)ij + (rαβ)ijl + γ k + (αγ)ik + (βγ)jk + (αβγ)ijk,
where blocks (rl), blocks × A ((ra)il), and blocks × A × B ((rαβ)ijl) are assumed to
contribute to the variation such that rl N 0, σ 2r , ðraÞil N 0, σ 2r × A ,
ðrαβÞijl N 0, σ 2rab , respectively, and εijkl experimental errors are distributed
as N(0, σ 2). This model uses the linear predictor ηijk to estimate the mean of the
observations μijk.
Link function: logit(π ijkl) = ηijkl
Table 6.18 below shows the data obtained from this experiment.
Table 6.19 presents the analysis of variance and shows the sources of variation
and degrees of freedom for this experimental design.
The following SAS GLIMMIX program allows a GLMM with a split-split plot
structure to be fitted in an RCBD with a binomial response.
234
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.19 Sources of variation and degrees of freedom for the randomized block design with an
arrangement of treatments under the split-split-plot structure
Sources of variation
Blocks
Factor A
Errora(Bloque*A)
Factor B
A* B
Errorb(A*B(Bloque))
Factor C
A*C
B*C
A*B* C
Error
Total
Degrees of freedom
r-1=2-1=1
a-1=3-1=2
(r - 1)(a - 1) = 2
b-1=2-1=1
(a - 1)(b - 1) = 2
a(b - 1)(r - 1) = 3 × 1 × 1 = 3
(c - 1) = 2 - 1 = 1
(3 - 1)(2 - 1) = 2
(b - 1)(c - 1) = 1
(a - 1)(b - 1)(c - 1) = 2
ab(c - 1)(r - 1) = 3 × 2 × 1 × 1 = 6
r × a × b × c - 1 = 2 × 3 × 2 × 2 - 1 = 23
proc GLIMMIX data=germ nobound method=laplace;
class Block A B C;
model Y/N = A|B|C/dist=binomial link=logit;
random block block*A block*A block*A*B;
lsmeans A|B|C/lines ilink;
run;
Part of the output is shown in Table 6.20. The value of the conditional statistic
Pearson’ chi - square/DF = 1.81 (part (a)) indicates that there is an overdispersion in
the dataset since these values are greater than 1. The estimated variance components
tabulated in part (b) correspond to blocks, blocks × factor A, and blocks × factor A × factor B, which are σ 2r = 0:0752, σ 2rA = 0:088, and σ 2rab = 0:0425, respectively. The type III tests of fixed effects are shown in part (c). Here, we see that
the test of equality of treatments is not significant for factors A and B and the
interaction AB (A, P = 0.1917, B, P = 0.0897; AB, P = 0.6262), whereas for factor
C and the interactions AC, BC, and ABC, it is significant at a level of 5%.
Since there is overdispersion in the dataset, the binomial distribution does not
provide a good fit for the dataset (Pearson’s chi - square/DF = 1.81). An alternative
to model this dataset could be the beta distribution. Under this assumption, let the
y
response variable be pijk = Nijkijk , the proportion of seeds that germinated, then pijk is
assumed to have a beta distribution rather than a binomial distribution for the success
count yijk out of a total of Nijk Bernoulli trials.
The components of the model are listed below:
Distribution: pijk j rl, (ra)il, (rαβ)ijl ~ Beta(π ijk, ϕ), with ϕ as the scale parameter.
rl N 0, σ 2r , ðraÞrk N 0, σ 2rA , ðrαβÞijl N 0, σ 2rab
Linear predictor:
ηijk = η + αi + rl + (rα)il + βj + (αβ)ij + (rαβ)ijl + γ k + (αγ)ik + (βγ)jk + (αβγ)ijk
Link function: logit π ijk = logit
π ijk
1 - π ijk
= ηijk
6.5
A Split-Split Plot in an RCBD:- In Vitro Germination of Seeds
235
Table 6.20 Results of the analysis of variance of the RCBD in the split-split plot under the
binomial distribution
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Estimate
Bloque
0.07521
Bloque*A
0.08847
Bloque*A*B
0.02205
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
A
2
2
B
1
3
A*B
2
3
C
1
6
A*C
2
6
B*C
1
6
A*B*C
2
6
146.19
43.49
1.81
Standard error
0.1180
0.09319
0.04258
F-value
4.22
6.12
0.55
65.73
11.68
29.38
31.69
Pr > F
0.1917
0.0897
0.6262
0.0002
0.0085
0.0016
0.0006
The following SAS commands fit a GLMM on a split-split plot in a randomized
complete block design assuming a beta distribution for the response variable.
proc glimmix data=germ nobound method=laplace;
class BlockABC ;
model p = A|B|C/dist=beta ;
random block block*A block*A*B;/*intercept A /subject=block*/;
lsmeans A|B|C/lines ilink;
run;
Part of the results is listed in Table 6.21 under a beta distribution. The value of the
fit statistic for the conditional model tabulated in (a) (Pearson’s chi - square/
DF = 1.01) indicates that overdispersion has been removed and that the
beta distribution is a good model to fit the dataset. Part (b) shows the variance
component
estimates
for
blocks,
blockxA,
and
blockxAxB
σ^2r = - 0:157, σ 2rA = - 0:05558, and σ 2rab = - 0:227, respectively and the value
^ = 19:2789 . According to the type III tests of
of the estimated scale parameter ϕ
fixed effects in part (c), the main effect of factor C (P = 0.0128) and interaction
A×B×C (P = 0.0424) are statistically significant at a level of 5%.
The estimates of the interactions are shown in Table 6.22 on the model scale
under the “Estimate” column and as probabilities on the data scale under the “Mean”
column with its corresponding standard errors under the “Standard error mean”
column.
236
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.21 Results of the analysis of variance of the RCBD in the split-split plot structure under
the beta distribution
(a) Fit statistics for conditional distribution
-2 Log L (p | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Estimate
Bloque
-0.1570
Bloque*A
-0.05558
Bloque*A*B
-0.2270
Scale
19.2789
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
A
2
2
B
1
2
A*B
2
2
C
1
4
A*C
2
4
B*C
1
4
A*B*C
2
4
-37.51
21.31
1.01
Standard error
.
.
.
5.8703
F-value
1.21
0.00
1.08
18.34
1.50
6.56
7.72
Pr > F
0.4521
0.9687
0.4799
0.0128
0.3257
0.0626
0.0424
Table 6.22 Estimated least mean squares on the model scale (“Estimate” column) and the data
scale (“Mean” column)
A*B*C least squares means
Standard
A B C Estimate error
1 1 1 -0.3769 0.3194
1 1 2
0.9506 0.3445
1 2 1
0.1721 0.3147
1 2 2
0.7010 0.3308
2 1 1 -0.6521 0.3296
2 1 2
2.9148 0.8071
2 2 1
0.7430 0.4699
2 2 2
0.4056 0.4515
3 1 1
0.2695 0.3161
3 1 2
0.2752 0.3163
3 2 1
0.1236 0.3143
3 2 2
1.1726 0.3614
DF
4
4
4
4
4
4
4
4
4
4
4
4
tvalue
-1.18
2.76
0.55
2.12
-1.98
3.61
1.58
0.90
0.85
0.87
0.39
3.24
Pr > |t|
0.3034
0.0509
0.6135
0.1014
0.1190
0.0225
0.1890
0.4198
0.4419
0.4334
0.7143
0.0315
Mean
0.4069
0.7212
0.5429
0.6684
0.3425
0.9486
0.6776
0.6000
0.5670
0.5684
0.5309
0.7636
Standard error
mean
0.07709
0.06927
0.07810
0.07331
0.07422
0.03937
0.1026
0.1084
0.07761
0.07759
0.07827
0.06523
Alternative Link Functions for Binomial Data
Average germination rate
6.6
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
237
a
ab
ab
abc
a..
abc
bc
bc
bc
bc
C1
C2
C1
c
c
C1
C2
C1
B1
C2
C1
B2
A1
C2
C1
B1
C2
B2
B1
A2
C2
B2
A3
Interaction
Fig. 6.7 The average seed germination rate
The simple effects of factors show that the best combination of factor levels was
A2*B1*C2, showing the highest seed germination proportion followed by the
combination of factors A1*B1*C2, A3*B2*C2, and lower proportion, which were
observed in the combination of factors A1*B2*C2, A2*B2*C1 and A2*B2*C2
(Fig. 6.7). Finally, the combination of the factor levels A2 × B1 × C1 showed the
lowest proportion of seed germination.
6.6
Alternative Link Functions for Binomial Data
In previous chapters, we used proc GLIMMIX with binomial data and, by default, it
works with the link function “logit. ” However, in certain applications with binomial
data, other link functions are acceptable, either because they make it easier to
interpret or because for certain binomial datasets, the link function “logit” cannot
accurately model the data and, as a result, produce biased (misleading) results. In this
section, we consider two alternative link functions to the logit for binomial data: the
link “probit” and the complementary log-log link.
The probit model is also used to model dichotomous (Bernoulli) or binomial (sum
of Bernoulli trials) responses. For this model, the link function, called the probit link,
uses the inverse of the cumulative distribution function of a standard normal
distribution to transform probabilities to the standard normal variable. That is,
1 2
z
Φ-1(π i) = ηi, which implies that π i = Φ(ηi), where ΦðZ Þ = - 1 p12π e - 2t dt.
The use of the probit regression model dates back to Bliss (1934). Bliss was
interested in finding an effective pesticide to control insects that fed on grape leaves.
238
6 Generalized Linear Mixed Models for Proportions and Percentages
He discovered that the relationship between the response and a dose of pesticide was
sigmoid, and he applied the probit link function to transform the dose–response
curve from a sigmoid to a linear relationship.
The complementary function log - log defined as ηi = log (- log (1 - π i)),
η
whose inverse is π i = 1 - e - e i , is useful for data in which most of the probabilities
are near zero or near one. For small values of π i, the log-log transformation produces
results highly similar to those produced when using a logit link. As the probability
increases, the transformation approaches infinity more slowly than the probit or logit
model.
6.6.1
Probit Link: A Split-Split Plot in an RCBD
with a Binomial Response
This example takes the dataset of the split-split plot in an RCBD (Exercise 6.8.5). In
this example, the data were modeled using the function “logit.” In this exercise, we
will fit the dataset using the link function “probit, ” and we will compare and contrast
the results using a logit link. The components of the GLMM are identical to those in
Example 6.5, except for the link function. That is, we replace:
Link function: logit π ijk = logit
π ijk
1 - π ijk
= ηijk by Φ-1(π ijk) = ηijk.
The following GLIMMIX syntax implements the fitting of the binomial data
using the link function “probit. ”
proc glimmix data=germ nobound method=laplace;
class Block A B C;
model Y/N = A|B|C/link=probit;
random block block*A block*A*B;
lsmeans A|B|C/lines ilink;
run;
Table 6.23 shows part of the results under the binomial distribution with the
“probit” link function. In parts (a) and (b), we see the mean squared error and
variance component estimates for blocks, whole plot, subplot, and sub-subplot,
where it can be observed that these values are positive and not negative, as the
ones obtained with the link function “logit. ” Since the variance components are
positive, this analysis makes more sense than the one based on the logit link.
The type III tests of fixed effects are tabulated in part (c) of Table 6.23; the main
effects of factors A and B and the interactions A*B, A*C, and B*C are not significant
in both link functions, whereas the main effect of factor C and the interaction A*B*C
are statistically significant under the “probit” link.
The estimated probabilities π^ijk and their respective standard errors are
presented in Table 6.24 for each of the combinations of the three factors, which
6.6
Alternative Link Functions for Binomial Data
239
Table 6.23 Results of the analysis of variance of the RCBD in the split-split plot structure under
the binomial distribution using the “probit” link
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
146.43
43.01
1.09
Pearson’s chi-square/DF CME = σ^2
(b) Covariance parameter estimates
Cov Parm
Estimate
0.02411
Block σ^2block
Block*A σ^2block × A
Block*A*B σ^2block × AB
Standard error
0.03707
0.02128
0.02830
0.01617
0.01896
(c) Type III tests of fixed effects
Effect
A
B
A*B
C
A*C
B*C
A*B*C
Num DF
2
1
2
1
2
1
2
Den DF
2
3
3
6
6
6
6
F-value
5.49
4.17
0.36
67.13
12.34
29.16
33.93
Probit
Pr > F
0.1541
0.1339
0.7226
0.0002
0.0075
0.0017
0.0005
Logit
Pr > F
0.4521
0.9687
0.4799
0.0128
0.3257
0.0626
0.0424
are very similar in both link functions. However, the average standard error
is slightly higher with the “logit” link function ðstandar:error:meanlogit = 0:0711Þ
compared to the “probit” link ðstandar:errormeanprobit = 0:0693Þ.
6.6.2
Complementary Log-Log Link Function: A Split Plot
in an RCBD with a Binomial Response
Researchers studied three different micro-minerals (A, B, and C) on the attachment
of explants of a commercial culture. In this vein, micro-mineral A was tested at three
levels (i = 1, 2, and 3), and micro-minerals B and C at two levels ( j, k = 1,2 and).
The combination of the different levels yielded a total of 12 combinations. Since the
researchers wanted to study factor C with greater precision, a split-plot treatment
structure was designed in which micro-minerals A and B were placed in the whole
plot (a large plot) and micro-mineral C in the subplot (a small plot). Treatment factor
combinations were placed in an RCBD manner (r = 1, 2). The outcome of interest
was the number of live plants (yijkr) out of the total number of plants growing in the
240
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.24 Means and standard errors using the probit and logit link functions
A*B*C least squares means
Probit
A
B
C
Mean
1
1
1
0.1543
1
1
2
0.3723
1
2
1
0.2724
1
2
2
0.2954
2
1
1
0.1023
2
1
2
0.8255
2
2
1
0.5684
2
2
2
0.5529
3
1
1
0.2844
3
1
2
0.2751
3
2
1
0.2568
3
2
2
0.4612
Standard error mean
0.05050
0.08296
0.06746
0.07798
0.03805
0.06338
0.08306
0.08327
0.07196
0.06868
0.06452
0.08017
Logit
Mean
0.1494
0.3780
0.2694
0.2953
0.09593
0.8292
0.5703
0.5530
0.2844
0.2733
0.2563
0.4608
Standard error mean
0.04796
0.08767
0.06896
0.08053
0.03409
0.06135
0.08845
0.08847
0.07418
0.07041
0.06589
0.08553
unit (nijkr). The data can be referred to in the Appendix (Data: Commercial crop
explant attachment).
The GLMM for this experiment is described below (log-log data):
Distribution: yijkl j rl, r(aβ)ijl~Binomial(Nijk, π ijk)
r l N 0, σ 2r , r ðaβÞijl N 0, σ 2rab ,
Linear predictor: ηijkl = η + rl + αi + βj + (αβ)ijl + r(αβ)il + γ k + (αγ)ik + (βγ)jk + (αβγ)ijk,
i + βj + (αβ)ijl + r(αβ)il + γk + (αγ)ik + (βγ)jk + (αβγ)ijk, where blocks (rl) and blocks
x (A x B) ((r(aβ))ijl) are assumed to contribute to the variation such that rl N 0, σ 2r and r ðaβÞijl N 0, σ 2rab , respectively.
Link function: log - log (π ijkl) = ηijkl
The following GLIMMIX code adjusts the binomial proportions with a complementary link function log - log in an RCBD manner.
proc glimmix data=spp nobound method=laplace;
class block A B C;
model y/n = A|B|C/link=ccll;
random block block(A*B);
lsmeans A|B|C/lines ilink;
run;
The “link = ccll” option specifies that “proc GLIMMIX” will fit the model using
the complementary (log - log) link function. The “lsmeans A|B|C/lines ilink”
command calls for estimation of the linear predictors ηijk, whereas the “lines” and
“ilink” options provide the comparison between the linear predictors and their
inverse. Part of the output is shown below. Table 6.25 shows the variance component estimates of blocks and blocks (A×B) using alternative link functions. Under
6.6
Alternative Link Functions for Binomial Data
241
Table 6.25 Variance component estimates using the same distribution but a different link function
Covariance parameter estimates
Log – log
Standard
Cov Parm
Estimate error
Block
0.05808 0.07112
Block
0.05065 0.03121
(A*B)
Logit
Estimate
0.08144
0.09203
Probit
Standard
error
0.1042
0.05754
Estimate
0.02676
0.03374
Standard
error
0.03494
0.02111
Table 6.26 Type III tests of fixed effects using the same distribution but with a different link
function
Type III tests of fixed effects
Effect
A
B
A*B
C
A*C
B*C
A*B*C
Num DF
2
1
2
1
2
1
2
Den DF
5
5
5
6
6
6
6
Table 6.27 Fit statistics
using the same distribution but
a different link function
Log - log
F-value Pr > F
6.27
0.0434
4.85
0.0789
0.65
0.5613
68.84
0.0002
11.94
0.0081
27.51
0.0019
32.44
0.0006
Logit
F-value
7.44
3.13
0.28
65.29
11.53
28.88
32.36
Pr > F
0.0318
0.1370
0.7693
0.0002
0.0088
0.0017
0.0006
Covariance parameter estimates
Log - log
-2 Log likelihood
164.85
AIC (smaller is better)
192.85
AICC (smaller is better)
239.51
BIC (smaller is better)
174.55
CAIC (smaller is better)
188.55
HQIC (smaller is better)
154.58
Probit
F-value
8.17
2.81
0.24
66.70
12.12
28.77
33.93
Pr > F
0.0266
0.1543
0.7971
0.0002
0.0078
0.0017
0.0005
Logit
172.57
200.57
247.24
182.27
196.27
162.31
Probit
170.88
198.88
245.55
180.59
194.59
160.62
the link “probit,” the variance components are smaller compared to those obtained
with the link functions “log – log” and “logit.”
The values of the hypothesis tests for the fixed effects, both main effects and
interactions, are shown in Table 6.26. The three link functions behave similarly.
One tool that might be useful in choosing which link function provides a better fit,
or which best describes the variability of a dataset, is the model fit statistics. The fit
statistics indicate that the model with the complementary “log - log” link function
provides the best fit (Table 6.27).
Table 6.28 shows the maximum likelihood estimators π^ijk for each of the link
functions and the combination of factor levels, and it can be verified that they
provide very similar estimates. It is important to mention that the correct
242
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.28 Means and standard errors using the same distribution but with a different link function
A*B*C least squares means
Log - log
Standard error
A B C Mean
mean
1 1 1 0.1494 0.04259
1 1 2 0.3776 0.08554
1 2 1 0.2661 0.06257
1 2 2 0.3001 0.07718
2 1 1 0.1020 0.03079
2 1 2 0.8389 0.08212
2 2 1 0.5558 0.09578
2 2 2 0.5578 0.09596
3 1 1 0.2770 0.06780
3 1 2 0.2782 0.06574
3 2 1 0.2555 0.05987
3 2 2 0.4599 0.08735
Logit
Mean
0.1513
0.3727
0.2706
0.2993
0.1023
0.8188
0.5733
0.5560
0.2805
0.2779
0.2561
0.4610
Probit
Standard error
mean
0.04732
0.08510
0.06744
0.07951
0.03451
0.06189
0.08633
0.08635
0.07192
0.06929
0.06416
0.08331
Mean
0.1547
0.3696
0.2737
0.2980
0.1047
0.8196
0.5700
0.5546
0.2827
0.2778
0.2569
0.4609
Standard error
mean
0.05030
0.08223
0.06718
0.07789
0.03829
0.06375
0.08251
0.08273
0.07131
0.06855
0.06410
0.07965
specification of the linear predictor as well as the distribution of the response variable
are the most important elements for obtaining a good fit.
6.7
Percentages
In this section, we consider proportions that have been calculated from discrete
counts, for example, the number of infected plants in treatment i of total Ni plants
that are likely to have a binomial distribution. This class of models allows the
response to arise from different distributions and probabilities.
6.7.1
RCBD: Dead Aphid Rate
An experiment was designed to study the effect of conidial density on the transmission of a fungus that attacks aphids. Aphid carcasses killed by the fungus, and from
which the fungus released spores, were placed on bean plants at three densities
(A = 1, B = 5, or C = 10 carcasses per plant) to provide different doses of fungal
conidia. Densities were assigned to individual bean plants in a completely randomized design with six replicates. A total of 20 live uninfected (N ) aphids were placed
on each plant with a ladybug that was allowed to forage (feed on the bean plants) to
facilitate the transfer of conidia between the carcasses and the live aphids. For each
plant, the number of aphids infected with the fungus was counted (nij) and the
proportion of aphids infected with the fungus was calculated 7 days after the
6.7
Percentages
243
Table 6.29 Proportion of
infested aphids
Plant
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Table 6.30 Sources of variation and degrees of freedom
Sources of variation
Trt
Error
Total
Density
C
A
B
C
B
B
C
C
A
B
A
A
C
C
A
B
B
A
pij
0.34299
0.16659
0.47004
0.62481
0.21926
0.16659
0.47502
0.52747
0.41581
0.42556
0.19466
0.34299
0.677
0.76674
0.13124
0.58419
0.38225
0.28905
Degrees of freedom
t-1=2
t(r - 1) = 15
t × r - 1 = 17
inoculum was placed. The results shown below correspond to the proportion of
infected aphids calculated at each of the inoculum concentrations ( pij = nij/N;
N = 20) to each of the conidial concentrations (density) tested (Table 6.29).
The sources of variation and degrees of freedom for this experiment are shown in
Table 6.30.
The components of the GLMM having a beta response are listed below:
Distributions: pij j density(plant)i( j ) ~ Beta(π ij, ϕ)
2
densityðplantÞiðjÞ N 0, σ density
ðplantÞ
Linear predictor: ηij = μ + densityi + density(plant)i( j ); i = 1, 2, 3; j = 1, ⋯, 6
Link function: log
π ij
1 - π ij
= logit π ij = ηij
The following GLIMMIX program fits a GLMM in a completely randomized
design with a beta distribution. Here, density is conc_ino.
proc glimmix data=thumbs nobound method=laplace;
class plant conc_ino;
model p = conc_ino /dist=beta link=logit;
random conc_ino(plant);
244
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.31 Results of the
analysis of variance
(a) Fit statistics for conditional distribution
-2 Log L (P | r. effects)
-24.13
Pearson’s chi-square
18.45
Pearson’s chi-square/DF
1.02
(b) Covariance parameter estimates
Cov Parm
Estimate
Standard error
Conc_Ino (Planta)
-0.1833
.
Scale
12.9999
4.1954
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Conc_Ino
2
15
8.25
0.0038
Table 6.32 Means and standard errors on the model scale and the data scale
Conc_Ino least squares means
Conc_Ino
A
B
C
Estimate
-1.0340
-0.5282
0.2775
Standard error
0.2438
0.2246
0.2197
DF
15
15
15
tvalue
-4.24
-2.35
1.26
Pr > |t|
0.0007
0.0328
0.2259
Mean
0.2623
0.3709
0.5689
Standard error mean
0.04717
0.05241
0.05388
lsmeans conc_ino/lines ilink;
run;
Part of the results is shown in Table 6.31. The value of the conditional fit statistic
in part (a), Pearson’s chi - square/DF = 1.02, indicates that there is no
overdispersion in the data and that the beta distribution is a good model for this
dataset. The estimated variance of the plants’ nested inoculum density is
^ = 12:999; both are
σ^2densityðplantÞ = - 0:1833 and the estimated scale parameter is ϕ
tabulated in part (b). In part (c) of the same table, the type III tests of fixed effects are
shown, indicating that the density (concentration) of the inoculum has a significant
effect (P = 0.0038) on the proportion of infected aphids with the fungus.
The values under the column “Estimates” are estimated mean proportions on the
model scale, whereas the column “Mean” shows the estimated mean proportions on
the data scale with their respective standard errors (Table 6.32). These estimates
where obtained with the “lsmeans” and “ilink” option.
Figure 6.8 shows a linear trend in the proportion of aphids infested as conidial
density increases. Conidia densities A and B showed statistically equal proportions
of infested aphids compared to density C. Finally, the highest proportion of infested
aphids was observed at density C.
6.7
Percentages
245
Proportion of infested aphids
0.7
0.525
0.35
0.175
0.
A
B
C
Conidial density
Fig. 6.8 Proportion of aphids infected at different conidia concentration densities
6.7.2
RCBD: Percentage of Quality Malt
An agro-industrial engineer is interested in studying the effect of germination time in
minutes (48, 96, and 144) on the percentage of quality malt obtained from six
sorghum varieties (sorghum bicolor): Gambella 1107, Macia, Meko, Red Swazi,
Teshale, and 76T1#23 (Bekele et al. 2012). The percentage of quality malt ( y) as a
function of both factors is shown in Table 6.33.
For this purpose, an RCBD was implemented with a treatment factorial structure
(variety × germination time). The statistical model to analyze the dataset is the
following:
Distributions: yijk j rk ~ Beta(π ijk, ϕ); i = 1, ⋯, 6; j, k = 1, 2, 3
rk N 0, σ 2block , where yijk is the kth percentage of malt quality observed at the ith
variety with the jth fermentation time.
Linear predictor: ηijk = μ + rk + αi + βj + (αβ)ij, where μ is the overall mean, αi is the
fixed effect due to variety i, βj is the fixed effect due to germination time j,
and (αβ)ij is the interaction effect between variety and germination time.
Link function: logit(π ijk) = ηijk
Table 6.34 shows the sources of variation and degrees of freedom for this
experiment.
The following GLIMMIX commands adjust a GLMM with a beta response.
proc glimmix data=malting nobound method=laplace;
class var_sorghum ger_time block;
model p = var_sorghum|ger_time/dist=beta link=logit;
random block;
lsmeans var_sorghum|ger_time/lines ilink;
run;
246
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.33 Percentage of quality malt as a function of both factors (variety and germination time)
Variety
Gambella
Gambella
Gambella
Macia
Macia
Macia
Meko
Meko
Meko
Red Swazi
Red Swazi
Red Swazi
Teshale
Teshale
Teshale
76T1#23
76T1#23
76T1#23
Gambella
Gambella
Gambella
Macia
Macia
Macia
Meko
Meko
Meko
Time
T1
T1
T1
T1
T1
T1
T1
T1
T1
T1
T1
T1
T1
T1
T1
T1
T1
T1
T2
T2
T2
T2
T2
T2
T2
T2
T2
Block
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
Table 6.34 Sources of variation and degrees of freedom
y
7.25
11.16
15.9
10.91
8.75
10.87
24.65
23.63
28.75
20.95
15.82
25.24
25.92
27.64
28.03
23.39
19.43
25.55
10.03
12.9
17.84
7.88
9.14
11.99
22.97
25.37
25.71
Variety
Red Swazi
Red Swazi
Red Swazi
Teshale
Teshale
Teshale
76 T1#23
76 T1#23
76 T1#23
Gambella
Gambella
Gambella
Macia
Macia
Macia
Meko
Meko
Meko
Red Swazi
Red Swazi
Red Swazi
Teshale
Teshale
Teshale
76 T1#23
76 T1#23
76 T1#23
Sources of variation
Blocks
Variety
Time_Germination
Variety*germ_time
Error
Total
Time
T2
T2
T2
T2
T2
T2
T2
T2
T2
T3
T3
T3
T3
T3
T3
T3
T3
T3
T3
T3
T3
T3
T3
T3
T3
T3
T3
Block
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
y
21
15.09
24.84
25.42
26.86
26.64
23.69
20.71
26.14
12.45
15.34
17.32
8.51
8.15
13.07
22.09
24.11
24.47
20.81
16.05
23.7
26.42
27.07
28.01
24.18
19.58
25.74
Degrees of freedom
r-1=3-1=2
a-1=6-1=5
b-1=3-1=2
(a - 1)(b - 1) = 10
(ab - 1)(r - 1) = 17 × 2 = 34
r × a × b - 1 = 54 - 1 = 53
Part of the results of the above program is shown in Table 6.35. In part (a), the
2
value of Pearson’s chi-square/DF is tabulated χdf = 0:92 , which indicates that the
beta distribution is a good distribution for modeling malt percentage since the t-value
of Pearson’s chi-square/DF is close to 1. The estimated variance due to blocks is
6.7
Percentages
Table 6.35 Results of the
analysis of variance of the
RCBD with a beta distribution
247
(a) Fit statistics for conditional distribution
-2 Log L (p | r. effects)
-280.89
Pearson’s chi-square
49.66
Pearson’s chi-square/DF
0.92
(b) Covariance parameter estimates
Cov Parm
Estimate
Standard error
Block
0.01210
0.01055
Scale
431.54
85.4922
(c) Type III tests of fixed effects
Num
Den
FDF
DF
value
Effect
Pr > F
Var_sorghum
5
34
106.51 <0.0001
Ger_time
2
34
0.26
0.7722
Var_sorghum*ger_time 10
34
1.08
0.4041
Table 6.36 Means and standard errors on the model scale and the data scale for sorghum varieties
Var_sorghum least squares means
Standard
Var_sorghum Estimate error
76 T1#23
-1.2011 0.07401
Gambella
-1.8898 0.07929
Macia
-2.2067 0.08295
Meko
-1.1201 0.07364
Red Swazi
-1.3685 0.07493
Teshale
-1.0025 0.07314
DF
34
34
34
34
34
34
t-value
-16.23
-23.83
-26.60
-15.21
-18.26
-13.71
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
0.2313
0.1313
0.09915
0.2460
0.2029
0.2685
Standard error
mean
0.01316
0.009042
0.007409
0.01366
0.01212
0.01436
^ = 431 (part (b)), whereas the
σ^2block = 0:012 and the estimated scale parameter is ϕ
type III fixed effects hypothesis tests in part (c) show that sorghum variety has a
significant effect on malt quality percentage (P = 0.0001).
The least squares means on the model scale and the data scale for the factor
variety are listed under the columns “Estimate” and “Mean” with their respective
standard errors “Standard error” in Table 6.36.
Figure 6.9 shows that Teshale produced the highest average malt percentage
(0.2685 ± 0.01436), followed by the varieties 76 T1#23 and Meco
(0.2313 ± 0.01316,0.246 ± 0.01366), whereas the variety Macia produced the
lowest malt percentage (0.09915 ± 0.0074).
6.7.3
A Split Plot in an RCBD: Cockroach Mortality
(Blattella germanica)
An entomologist is interested in testing six isolates of insect pathogenic fungi: five
obtained from different hosts and one already known isolate (Control) of a fungus
248
6
Generalized Linear Mixed Models for Proportions and Percentages
Average malt percentage
0.3
0.225
0.15
0.075
0.
76T1#23
Gambella
Macia
Meko
RedSwazi
Teshale
Variety
Fig. 6.9 Percentage of quality malt of bicolor sorghum varieties
Table 6.37 Analysis of variance with sources of variation and degrees of freedom for this
experiment
Sources of variation
Blocks
Isolation
Block (insulation)
Age
Isolation*age
Error
Total
Degrees of freedom
r - s1 = 2 - 1 = 1
a-1=6-1=5
a(r - 1) = 6
b-1=3-1=2
(a - 1)(b - 1) = 5 × 2 = 10
(a - 1)(b - 1)(r - 1) = 2 × 5 × 1 = 10
r × a × b - 1 = 2 × 6 × 3 - 1 = 35
with potential for biological control of a particular species of cockroaches. To do so,
the entomologist decides to test these fungal isolates on three different insect ages
(age1 = E1, age2 = E2, and age3 = E3). Each of the isolates was placed in a Petri
dish with 10 insects of a specific age. Each set (isolate–age) was randomly assigned
to two blocks (Appendix: Data: Cockroaches).
The analysis of variance table (Table 6.37) with the sources of variation and
degrees of freedom for this experiment is presented below. The response variable
(percentage mortality) for this experiment is assumed to have a beta distribution.
The components that describe the model of this experiment are listed below:
Distributions: yijk j rk, r(α)k(i)~Beta(π ijk, ϕ); i = 1, ⋯, 6; j = 1, 2, 3; k = 1, 2.
rk N 0, σ 2r , r ðαÞkðiÞ N 0, σ 2rðαÞ
Linear predictor: ηijk = μ + rk + αi + r(α)k(i) + βj + (αβ)ij
Link function: logit(π ijk) = ηijk
6.7
Percentages
Table 6.38 Results of the
analysis of variance of the
RCBD with a factorial structure in treatments
249
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
-74.53
Pearson’s chi-square
34.02
Pearson’s chi-square/DF
1.00
(b) Covariance parameter estimates
Cov Parm
Subject
Estimate
Standard error
Aislamiento
Block
-0.03125
.
Scale
24.1882
5.7925
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Isolation
5
6
16.48
0.0019
Age
2
10
30.01
<0.0001
Isolation*age
10
10
4.83
0.0102
The following GLIMMIX commands adjust a GLMM with a beta response.
proc glimmix nobound method=laplace;
class block Isolation Age;
model y = Isolation|Age/dist=beta link=logit;
random Isolation/subject=block;
lsmeans Insulation|Age/slice=Insulation lines ilink;
run;
Some of the outputs are listed below (Table 6.38). The conditional statistic
Pearson′s chi - square/DF = 1 indicates that the distribution used is appropriate
for these datasets (part (a)). The variance component estimates are tabulated in part
(b), and, for blocks, the estimate is σ^2r = - 0:03125 and the estimated scale param^ = 24:1882. The hypothesis test is in part (c) with type III fixed effects of
eter is ϕ
equality of means for type of isolation, age of the insect, and the interaction between
both factors. These outputs indicate that they have a significant effect on insect
mortality.
We see the expected proportions with their respective standard errors of both
factors on the data scale under the “Mean” column (Tables 6.39 and 6.40). These
values arise by applying the inverse link to estimates under “Estimate” on the model
scale. Table 6.39 shows the estimated average mortality probabilities for the isolates;
for example, for isolate A1, applying the inverse link to the linear predictor estimate
^η1: = 0:1722 we get π^1: = 1=1 þ e - 0:1722 = 0:5429. In this manner, we see that the
expected proportions for isolates 2 and 4 are π^2: = 0:6555 and π^4: = 0:5762, respectively, whereas for the control π^control: = 0:1157.
Regarding the age of the insect (Table 6.40), the expected average probability of
mortality was higher at age three (adults) with a higher mortality rate π^:3 = 0:6435,
whereas insects at age two (E2) had a higher resistance to the isolations, showing a
mortality of π^:2 = 0:2598.
In general, fungal isolates A1, A2, A3, and A4 showed an average mortality of
more than 75% for adult insects (E3), whereas isolates A1, A2, and A5 showed a
250
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.39 Means and standard errors on the model scale and the data scale for isolation
Isolate least squares means
Isolate
Estimate Standard error
A1
0.1722 0.1859
A2
0.6442 0.2100
A3
-0.1489 0.1952
A4
0.3073 0.2088
A5
-0.2023 0.1806
Control -2.0339 0.2418
DF
6
6
6
6
6
6
t-value
0.93
3.07
-0.76
1.47
-1.12
-8.41
Pr > |t|
0.3900
0.0220
0.4746
0.1915
0.3053
0.0002
Mean
0.5429
0.6557
0.4629
0.5762
0.4496
0.1157
Standard error mean
0.04614
0.04740
0.04853
0.05098
0.04468
0.02473
Table 6.40 Means and standard errors on the model scale and the data scale for insect age
Age least squares means
Age Estimate Standard error
E1
-0.1747 0.1310
E2
-1.0468 0.1374
E3
0.5908 0.1634
DF
t-value
-1.33
-7.62
3.61
Pr > |t|
0.2120
<0.0001
0.0047
Mean
0.4564
0.2598
0.6435
Standard error mean
0.03251
0.02643
0.03749
mortality rate of around 65% for cockroaches of age E1 (juvenile insects). On the
other hand, all isolates showed lower lethal effectiveness on insects of age E2
(Fig. 6.10).
6.7.4
A Split-Plot Design in an RCBD: Percentage Disease
Inhibition
A plant pathologist wishes to compare the response of two plant varieties to different
doses/amounts of a pesticide formulated to protect plants against a disease. Five
racks (blocks) were chosen to account for local variation within the greenhouse.
Each rack was divided into four sections or rooms and were randomly assigned one
of four pesticide levels to each rack. The four pesticide levels were 1, 2, 4, and 8 mg/
L. One plant of each variety was placed in each section of the rack. Of the two plant
varieties, one variety was susceptible, labeled S, and the other variety was resistant,
labeled R (Table 6.41). The response variable ( y) is the percentage of disease
inhibition in the plant.
The sources of variation and degrees of freedom for this experiment are shown in
Table 6.42.
Following the same reasoning used in the examples above, the components of the
GLMM with a beta response that models the observed disease inhibition proportion
( pijk) under dose i with variety j in block k are listed as follows:.
Distributions: yijk j rk, (rα)ik~Beta(π ijk, ϕ); i = 1, ⋯, 4; j = 1, 2; k = 1, ⋯, 5
rk N 0, σ 2r , ðrαÞik N 0, σ 2rA
6.7
Percentages
251
1
Average mortality rate
0.9
0.8
0.7
a
ab
ab
ab
abc
bc
bc
de
0.5
0.4
0.3
cd
cd
cd
0.6
ef
ef
ef
f
f
0.2
f
f
0.1
0
E1 E2 E3 E1 E2 E3 E1 E2 E3 E1 E2 E3 E1 E2 E3 E1 E2 E3
A1
A2
A3
A4
A5
Control
Isolate/Age
Fig. 6.10 Cockroach mortality percentage
Table 6.41 Percentage of inhibition
Block
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
Variety
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
Dose
1
1
1
1
1
2
2
2
2
2
4
4
4
4
4
8
8
8
8
8
y
15.7
23.1
15.9
20.8
24.5
25.1
29.2
29.7
28.6
26.6
27.9
29.7
24
29.7
29.6
23.8
31.2
21.8
23.3
23.9
Block
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
Variety
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
Dose
1
1
1
1
1
2
2
2
2
2
4
4
4
4
4
8
8
8
8
8
y
19.8
17.8
13.2
14.8
19.7
21.2
29.3
26
27.5
22
29.3
27.2
26
31.5
27.9
22.8
33
25.2
27.2
20.8
252
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.42 Sources of variation and degrees of freedom
Sources of variation
Blocks
Dose
Errora(Bloque*Dose)
Variety
Dose*variety
Errorb
Total
Degrees of freedom
r-1=5-1=4
a-1=4-1=3
(r - 1)(a - 1) = 12
b-1=2-1=1
(a - 1)(b - 1) = 3
a(b - 1)(r - 1) = 4 × 1 × 4 = 16
r × a × b - 1 = 5 × 4 × 2 - 1 = 39
Linear predictor: ηijk = μ + rk + αi + (rα)ik + βj + (αβ)ij, where rk is the random block
effect, αi is the fixed dose effect, βj is the fixed variety effect, (rα)ik is the random
effect due to block by dose interaction, and (αβ)ij is the interaction of fixed effects
due to dose variety.
Link function: logit(π ijk) = ηijk
The following GLIMMIX commands adjust a GLMM.
proc glimmix nobound method=laplace;
class Variety dose block;
model y = dose variety dose*variety /dist=beta link=logit;
random Block Block*dose;
contrast 'Linear dose' dose -3 -1 1 3;
contrast 'Quadratic dose' dose 1 -1 -1 -1 1;
contrast 'dose Cubic' dose -1 3 -3 1;
lsmeans variety|dose / slice=(variety dose) lines ilink;
ods output lsmeans=dose_means;
run;
The “contrast” command in the program can perform a hypothesis testing to see
what trend (linear, quadratic, or cubic) the “dose” factor has on the percentage of
disease inhibition. Part of the output is shown in Table 6.43. The value of the
conditional goodness-of-fit statistic Pearson’s chi - square/DF= 0.59 indicates
that we have no evidence of overdispersion, and, therefore, the beta distribution is
adequate to model this dataset (part (a)). The variance component estimates in part
(b) for block and block × dose are σ^2r = 0:004898 and σ^2r ∙ dose = 0:002372, respectively. Finally, the F-value provides sufficient statistical evidence of the effect of
dose on disease decline in plants (P = 0.0001), whereas the effect of variety and dose
× variety do not provide sufficient evidence.
Table 6.44 shows the polynomial contrasts for the effect of “dose,” which
indicate that there is a significant quadratic effect on the percentage of disease
inhibition.
The inhibition percentage has almost a linear trend as the dose increases from 1 to
4 ml/L in both varieties, but when the dose is higher than 4 ml/L, the inhibition of the
disease decreases in both varieties (Fig. 6.11).
6.7
Percentages
253
Table 6.43 Results of the
analysis of variance
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
-184.32
Pearson’s chi-square
23.63
Pearson’s chi-square/DF
0.59
(b) Covariance parameter estimates
Cov Parm
Estimate
Standard error
Block
0.004898
.
Block*dose
0.002372
0.007513
Scale
205.52
67.7447
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Dose
3
12
17.67
0.0001
Variety
1
16
1.74
0.2057
Dose*variety
3
16
1.22
0.3337
Table 6.44 Polynomial
contrasts
Contrasts
Label
Linear dose
Quadratic dose
Cubic dose
6.7.5
Num DF
1
1
1
Den DF
12
12
12
F-value
25.48
30.93
0.30
Pr > F
0.0003
0.0001
0.5948
Randomized Complete Block Design with a Binomial
Response with Multiple Variance Components
The dataset corresponds to an experiment implemented by Madden and Hughes
(1995) on the incidence of the disease caused by the fungus Plasmopara viticola on
grape plants (Vitis labrusca). Six different treatments in a randomized block design
(b = 3) were tested, where treatment 1 was the control, to study the disease with
three grape plants (v = 3). On a single date in autumn, five sprouts were (r = 5)
randomly selected from each of the three grape plants and the number of leaves with
at least one mildew lesion was counted (m) out of a total n leaves. The number of
leaves per shoot ranged from 7 to 21. The data for this experiment can be found in
the Appendix (Data: Disease incidence on grape plants).
The statistical model that could describe the incidence of disease in this experiment, if the response variable pijkl were treated as a normal variable, would be as
described below:
pijkl = η þ τi þ bj þ ðbvÞjk þ ðbvr Þjkl þ εijkl
i = 1, 2, . . . , 6; j = 1, 2, 3; k = 1, 2, ::, 3; l = 1, 2, . . . , 5
254
6
Generalized Linear Mixed Models for Proportions and Percentages
Inhibition porcentage
0.35
0.3
0.25
0.2
0.15
0.1
1
2
3
4
5
6
7
8
Dose (ml/L)
r
s
Fig. 6.11 Percentage of disease inhibition in both varieties
where pijkl is the ijkl proportion of diseased leaves, η is the intercept, τi is the fixed
treatment effect i, bj is the random effect of blocks assuming bj N 0, σ 2block , (bv)jk
is the block–plant random effect assuming ðbvÞjk N 0, σ 2block × plant , (bvr)jkl
is
N
the
random
effect
2
0, σ block
× plant × sprout
due
to
block–plant–sprouts
assuming
ðbvrÞjkl , and εijkl is the experimental error assuming εijkl~N(0, σ 2).
For the disease incidence data, the assumption of a normal distribution for pijkl is
not recommended. A good starting point for the analysis is to assume that the
observed number of diseased leaves in the sprouts (yijkl) follows a binomial distribution with parameter π ijkl and nijkl, the total number of leaves on the sprout.
Therefore, the components of the GLMM with a binomial distribution in the
response variable are as follows:
Distribution: pijkl j bj, (bv)jk, (bvr)jkl ~ binomial(π ijkl, nijkl)
bj N 0, σ 2block ,ðbvÞjk N 0, σ 2block × plant , ðbvrÞjkl N 0, σ 2block × plant × sprout
Linear predictor: ηijkl = η + τi + bj + (bv)jk + (bvr)jkl.
Link function: logit(π ijkl) = ηijkl
The following GLIMMIX syntax fits a GLMM with a binomial response.
proc glimmix method=laplace nobound;
class v r b t;
model m/n = t /dist=bin;
random intercept v v*r/subject=b;
lsmeans t/lines ilink;
run;
6.7
Percentages
Table 6.45 Results of the
analysis of variance under the
binomial distribution
255
(a) Fit statistics
-2 Log likelihood
723.17
AIC (smaller is better)
741.17
AICC (smaller is better)
741.87
BIC (smaller is better)
733.06
CAIC (smaller is better)
742.06
HQIC (smaller is better)
724.87
(b) Fit statistics for conditional distribution
-2 Log L (m | r. effects)
665.02
Pearson’s chi-square
398.21
Pearson’s chi-square/DF
1.47
(c) Covariance parameter estimates
Cov Parm
Subject
Estimate
Standard error
Intercept
b
-0.00408
.
V
b
0.01917
.
v*r
b
0.1960
.
(d) Type III tests of fixed effects
Den DF
F-value
Pr > F
Effect
Num DF
t
2
220
1837.99
<0.0001
Part of the results based on the aforementioned model is shown in Table 6.45. By
default, proc GLIMMIX provides the fit statistics useful for selecting the best model
from a group of models (part (a)).
In addition to accuracy considerations, the Laplace (or quadrature) analysis
allows us to obtain the “conditional distribution fit statistics,” specifically
Pearson’s χ 2/df. Recall that this statistic helps assess the goodness of fit of the
model. If the value of χ 2/df ≫ 1 is an indicator that there is overdispersion in the
dataset, then this may be because the linear predictor is incomplete or the assumed
distribution is not suitable (mis-specified) for this dataset. In part (b), we can see that
the value of the conditional distribution statistic of Pearson’s χ 2/df = 1.47. This value
indicates that we have evidence of overdispersion. The variance component estimates due to block, block × plant, and block × plant × sprout are tabulated in part (c),
whereas the type III tests of fixed effects (part (d)) indicate that there is a significant
difference (P < 0.0001) between treatments.
Since there is overdispersion in the data in the binomial model, an alternative
distribution is the beta distribution. The components of the GLMM are as follows:
Distribution: pijkl j bj, (bv)jk, (bvr)jkl~beta(π ijkl, ϕ);
2
2
2
bj N 0, σ block
,ðbvÞjk N 0, σ block
× plant , ðbvrÞjkl N 0, σ block × plant × sprout
Linear predictor: ηijkl = η + τi + bj + (bv)jk + (bvr)jkl
Link function: logit(π ijkl) = ηijkl.
256
6
Table 6.46 Results of the
analysis of variance under the
beta distribution
Generalized Linear Mixed Models for Proportions and Percentages
(a) Fit statistics
-2 Log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
(b) Fit statistics for conditional distribution
-2 Log L (m | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(c) Covariance parameter estimates
Cov Parm
Subject
Estimate
Intercept
b
-0.6308
-0.2215
V
b
v*r
b
-0.1843
^
9.8397
Scale ϕ
(d) Type III tests of fixed effects
Effect
Num DF
Den DF
t
2
220
-231.10
-211.10
-209.30
-220.11
-210.11
-229.22
-231.10
136.55
1.07
Standard error
.
.
.
1.1926
F-value
1837.99
Pr > F
<0.0001
The following SAS commands adjust an GLMM under a beta distribution.
proc GLIMMIX method=laplace nobound;
class v r b t;
model pct = t /dist=beta link=logit;
random intercept v v*r/subject=b;
lsmeans t/lines ilink;
run;
Some of the outputs are shown below. Table 6.46 shows that the values of the fit
statistics, as well as the conditional distribution statistics (parts (a) and (b)), are much
smaller than when the binomial distribution was used.
This indicates that the beta distribution is more appropriate for the dataset, as
the value of Pearson’s statistic is χ 2/df = 1.03, indicating that the problem of
overdispersion was almost totally controlled. The variance component estimates as
^ are tabulated in part (c). Similar to the
well as the estimated scale parameter ϕ
previous analysis, the type III tests of fixed effects indicate that there is a highly
significant difference (part (d)) in treatments on the average proportion of leaves
with fungal disease.
The least mean squares (means) on the model scale (column “Estimate”) and on
the data scale (column “Mean”) are tabulated in Table 6.47. The results indicate that
6.8
Exercises
257
Table 6.47 Estimated means (least squares means) on the model scale and on the data scale
Least squares means
t
Estimate
Standard error
1
0.7223 0.09989
2 -1.7482 0.1543
3 -2.0178 0.2214
4 -1.9358 0.1873
5 -1.7887 0.2173
6 -1.5360 0.1665
Table 6.48 Mean comparison (LSD method)
DF
83
83
83
83
83
83
t-value
7.23
-11.33
-9.11
-10.34
-8.23
-9.23
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
0.6731
0.1483
0.1174
0.1261
0.1432
0.1771
Standard error mean
0.02198
0.01949
0.02294
0.02064
0.02667
0.02427
T grouping of t least squares means (α = 0.05)
LS means with the same letter are not significantly different
t
Estimate
1
0.7223
A
6
-1.5360
B
B
2
-1.7482
B
B
5
-1.7887
B
B
4
-1.9358
B
B
3
-2.0178
B
all proposed treatments in this study reduce the proportion of diseased leaves
compared to the control treatment (t = 1).
The mean comparison (LSD) obtained with the option “lines” indicates that the
proportion of diseased leaves in treatment one is statistically different from the rest
of the treatments (Table 6.48).
6.8
Exercises
Exercise 6.8.1 Seeds of a particular crop were stored at four different temperatures
(T1, T2, T3, and T4) under four different chemical concentrations (0, 0.1, 1.0, and 10).
To study the effects of temperature and chemical concentration, a completely
randomized experiment was conducted with a factorial treatment structure 4 × 4
and four replicates. For each of the 64 experimental units, 50 seeds were placed in a
dish and the number of seeds that germinated under standard conditions was
recorded. Germination data were obtained from Mead et al. (1993, p. 325)
(Table 6.49).
258
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.49 Seed germination experiment results
Temperature
T1
T2
T3
T4
Chemical concentration
0
0.1
9, 9, 3, 7
13, 12, 14, 15
19, 30, 21, 29
33, 32, 30, 26
7, 7, 2, 5
1, 2, 4, 4
4, 9, 3, 7
13, 6, 15, 7
1.0
21, 23, 24, 27
43, 40, 37, 41
8, 10, 6, 7
16, 13, 18, 19
40, 32, 43, 34
48, 48, 49, 48
3, 4, 8, 5
13, 18, 11, 16
Table 6.50 Results of the apple sprouts experiment
Density of inoculum
200
200
200
1000
1000
1000
5000
5000
5000
Cultivate
Jonagold
Golden delicious
Jonathan
Jonagold
Golden delicious
Jonathan
Jonagold
Golden delicious
Jonathan
Block 1
5/1
5/1
5/2
5/0
5/0
5/4
5/5
5/5
5/5
Block 2
5/2
5/0
5/2
5/2
5/0
5/4
5/5
5/4
5/0
Block 3
5/1
5/0
5/2
5/2
5/2
5/4
5/4
5/3
5/3
Block 4
5/0
5/0
5/0
5/4
5/0
5/0
5/5
5/5
5/5
The first number refers to the number of inoculations (n) and the second to the number of
inoculations that developed the gangrenous sore (Y)
(a) Write down an ANOVA table (sources of variation, degrees of freedom) for this
experiment.
(b) List all the components of the GLMM in (a).
(c) Analyze this dataset and summarize the relevant results.
Exercise 6.8.2 Data were obtained from an experiment in which separate sprouts of
apple trees were inoculated with macroconidia of the fungus Nectria galligena,
which causes apple cancer (canker gangrene). The experimental factors were inoculum density (three levels: 200, 1000, and 5000 macroconidia per ml) and variety
(three levels: Jonagold, Golden Delicious, and Jonathan). The experiment was
carried out in 4 randomized blocks with 12 plots. Each plot consisted of one sprout
on which five inoculations were made. The numbers of successful inoculations per
plot on day 17 after inoculation are shown in the table below (Table 6.50).
(a) Write down an ANOVA table (sources of variation, degrees of freedom) for this
experiment.
(b) List all the components of the GLMM from part (a).
(c) Analyze this dataset and summarize the relevant results.
(d) Is there is an extra-variation in the dataset? What alternative distribution do you
propose? Reanalyze the data and compare the results.
Exercise 6.8.3 This experiment concerns the germination efficiencies of protoplasts
obtained from plants of seven species of the genera Lycopersicon (tomato) and
6.8
Exercises
259
Table 6.51 Protoplast germination experiment results
Species
1
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
7
Isolation
1
2
3
4
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
4
1
8.9
3.1
2.1
2.5
0.2
1.8
6.6
1.8
1.5
2
11.4
2.9
2.3
21.5
18.7
11.5
4.6
2.4
1.6
3
2.5
2.6
2.9
2
6.3
2.7
1.9
2.9
0.9
1.6
7.5
1.5
3.2
2.3
11.3
3.8
4.4
25.5
20
13.1
3.4
2.4
1.1
4
2.5
2.7
3
3
10.5
4.1
1.4
2.6
0.5
1.6
5.4
1.9
1.1
2.8
14.4
4.7
4.8
18.1
4
11.5
2.7
2
1.6
4.1
2.5
2.9
3
5
1.5
2.6
0.6
6
7
8
9
10
2.6
1.2
2.6
0.4
2.8
2.7
2.8
2.7
5.3
1.7
1.3
2.6
13.7
5.1
4.9
22.2
5
1.3
1.8
3.2
6.5
1.5
1.2
2.2
6.3
5.8
5.9
5.6
1.6
2.5
1.4
2.4
1.2
2.8
1.8
2.4
2.7
5.8
3.2
4.7
5.6
4.2
3.3
4.5
16.2
3
2.5
1.3
4.4
2.7
2.7
3.1
10.1
4.1
3.6
1.6
2.8
2.3
2.7
17.2
3.1
3.2
1
3.3
2.6
2.6
16
10.5
2.6
0.8
4.5
1.4
1.3
3.3
2.5
0.8
3
2.7
2.2
3.2
Solanum (potato). For each species, three or four protoplast isolates were used and,
depending on the availability of the protoplasts, a variable number of plates was
carried out. Per plate, approximately 105 protoplasts were placed in a Petri dish, and,
after 4 weeks, the proportion of dividing protoplasts was recorded. The results in
percentages are listed below (Table 6.51).
(a) Write down an ANOVA table (sources of variation, degrees of freedom) for the
experimental design of this study.
(b) Write down a generalized linear mixed model base in (a), assuming a beta
distribution on the response variable.
(c) Implement an analysis of these data according to the linear predictor and model
in part (b). Summarize the relevant results.
Exercise 6.8.4 The data in this example are the results of a triangle test for 12 raters
tasting 10 pairs of coffee varieties (Table 6.52). The triangle test consisted of each
rater drinking three cups, one of one variety and two of the other. Each rater had
12 triangles for each pair of varieties, 2 for each of the following sequences: AAB,
ABA, BAA, ABB, BAB, and BBA. The answer is the correct variety identification
number appearing once. The experiment was conducted in two groups of six
260
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.52 Triangle test (G = group, Eval = panelist, PdV = variety pair, V_A = variety A;
V_B = variety B; Y = number of correct discriminations, n = number of trials)
G
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Eval
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
5
5
PdV
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
V_A
8
5
9
6
6
5
7
7
7
7
8
5
9
6
6
5
7
7
7
7
8
5
9
6
6
5
7
7
7
7
8
5
9
6
6
5
7
7
7
7
8
5
V_B
9
9
6
5
8
8
8
9
5
6
9
9
6
5
8
8
8
9
5
6
9
9
6
5
8
8
8
9
5
6
9
9
6
5
8
8
8
9
5
6
9
9
Y
2
11
9
6
8
9
6
8
11
5
5
8
8
9
10
11
8
9
8
7
4
9
9
11
8
10
3
7
10
9
7
10
7
8
7
8
7
6
10
7
6
10
n
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
G
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Eval
7
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
10
10
10
10
10
10
10
10
10
10
11
11
PdV
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
V_A
8
5
9
6
6
5
7
7
7
7
8
5
9
6
6
5
7
7
7
7
8
5
9
6
6
5
7
7
7
7
8
5
9
6
6
5
7
7
7
7
8
5
V_B
9
9
6
5
8
8
8
9
5
6
9
9
6
5
8
8
8
9
5
6
9
9
6
5
8
8
8
9
5
6
9
9
6
5
8
8
8
9
5
6
9
9
Y
4
12
7
9
10
5
9
9
7
5
2
10
8
9
8
9
4
6
10
7
3
11
11
8
8
11
5
4
11
8
7
9
5
11
5
10
7
8
6
9
7
9
n
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
(continued)
6.8
Exercises
261
Table 6.52 (continued)
G
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Eval
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
PdV
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
V_A
9
6
6
5
7
7
7
7
8
5
9
6
6
5
7
7
7
7
V_B
6
5
8
8
8
9
5
6
9
9
6
5
8
8
8
9
5
6
Y
4
8
6
7
8
9
9
8
3
9
6
9
7
10
7
7
8
9
n
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
G
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Eval
11
11
11
11
11
11
11
11
12
12
12
12
12
12
12
12
12
12
PdV
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
V_A
9
6
6
5
7
7
7
7
8
5
9
6
6
5
7
7
7
7
V_B
6
5
8
8
8
9
5
6
9
9
6
5
8
8
8
9
5
6
Y
6
10
5
10
8
6
9
9
6
7
7
7
8
11
9
9
10
9
n
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
evaluators, each with the aim of discriminating the abilities of the panelists for future
evaluations. The data for this example are shown below:
(a) Write down an ANOVA table (sources of variation, degrees of freedom) for this
experiment.
(b) List all the components of the GLMM according to part (a).
(c) Analyze this dataset and summarize the relevant results.
(d) Is there an extra-variation in the dataset? If so, what alternative distribution do
you propose? Reanalyze the data and compare the results.
Exercise 6.8.5 Several brewing techniques are used in the production of espresso
coffee. Among them, the most widespread are bar machines and single-dose pods,
designed in large numbers due to their commercial popularity. This experiment tries
to compare the foaming rate (Y, in percentage) effects of three different brewing
techniques on espresso quality (method 1 = bar machine (BM), method 2 = hyperespresso method (HIP), and method 3 = I-espresso system (IT)). Nine replicates per
method were carried out (Table 6.53).
(a) Write down an ANOVA table (sources of variation, degrees of freedom) for the
experimental design of this study.
(b) Describe the generalized linear mixed model in (a), assuming a beta distribution.
(c) Implement the analysis of these data according to the predictor and model in (b).
Summarize the relevant results.
262
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.53 Experimental
results of espresso coffee
Method
1
1
1
1
1
1
1
1
1
Table 6.54 Results of wheat
germination experiment in
pots. Number of seeds that did
not germinate out of 50
Treatments
1
A
10
B
8
C
5
D
1
Index
36.64
39.65
37.74
35.96
38.52
21.02
24.81
34.18
23.08
2
11
10
11
6
Method
2
2
2
2
2
2
2
2
2
3
8
3
2
4
Index
70.84
46.68
73.19
57.78
48.61
72.77
65.04
62.53
54.26
Method
3
3
3
3
3
3
3
3
3
Index
56.19
36.67
35.35
40.11
33.52
37.12
37.33
32.68
48.33
7
9
11
11
10
4
5
6
9
7
8
13
7
9
10
7
6
3
7
10
Exercise 6.8.6 The decision to adopt a particular scale for data involving small
integers is not an easy one because any analysis must be – to some extent – as
adequate as possible to obtain estimates with as little uncertainty as possible. As a
simple example of this type of data, consider the following results from a potted
wheat germination experiment (Table 6.54).
(a) Write down an ANOVA table (sources of variation, degrees of freedom) for this
experiment.
(b) List all components of the GLMM in (a), assuming a binomial response variable.
(c) Analyze this dataset and summarize the relevant results.
(d) Is there an extra-variation in the dataset? If so, reanalyze the data with an
alternative distribution. Summarize and compare your findings.
Exercise 6.8.7 A greenhouse experiment was carried out to investigate how a
disease spreads in two varieties of (agurkesyge) cucumber, which is supposed to
depend on the climate and the amount of fertilizers used for the two varieties. The
following data come from the Department of Plant Pathology. Two climates
were used: (1) change to day temperature 3 hours before sunrise and (2) normal
change to day temperature. Three amounts of fertilizer were applied, normal
(2.0 units), high (3.5 units), and very high (4.0 units). The two varieties were
Aminex and Dalibor. To have a better controlled experiment, the plants were
“standardized” to equally have as many leaves, and, then (on day 0, for example),
the plants were contaminated with the disease. Subsequently, 8 days after the plants
were contaminated, the amount of infection (in percentage) was recorded. From the
resulting infection curve, two measures were calculated (in a manner not specified
here), namely, the rate of spread of the disease (%) and the level of infection at the
6.8
Exercises
263
end of the disease period. The experiment was implemented in three blocks, each of
which consisted of two sections. Each section consisted of three plots, which were
divided into two subplots, each of which had six to eight plants. Thus, there were a
total of 36 subplots. The results were recorded for each subplot. The experimental
factors were randomly assigned to the different units as follows: two climates to the
two sections within each block, three amounts of fertilizer to the three plots within
each section, and, finally, the two varieties to the two subplots within each plot. The
data are shown below (Table 6.55).
(a)
(b)
(c)
(d)
Write down a statistical model of this experiment.
List all the components of the GLMM in (a).
Write down the null and alternative hypotheses associated with this experiment.
Construct an ANOVA table indicating the sources of variation and degrees of
freedom.
(e) Analyze the rate of disease spread to investigate the effect of different factors.
(f) Comment on the results obtained.
Exercise 6.8.8 This example is an experiment to identify damage to the uterus in
laboratory rodents after exposure to boric acid, a compound widely used in pesticides, pharmaceuticals, and other household products (Heindel et al. 1992). The
study design included four doses of boric acid. The compound was administered to
pregnant female mice during the first 17 days of gestation, and, then, the females
were sacrificed and their litters examined. The table below presents the resulting
trials for litters dying in utero (Y ) of the total number of trials conducted (N ) at each
of the four doses tested: d1 = 0{control}, d2 = 0.1, d3 = 0.2, and d3 = 0.4
(as percentage of boric acid in the diet) (Table 6.56).
(a) Write down an ANOVA table (sources of variation, degrees of freedom) for this
experiment.
(b) List all the components of the GLMM in (a).
(c) Analyze this dataset and summarize the relevant results.
(d) Is there an extra-variation in the dataset? If so, what alternative distribution do
you propose? Reanalyze the data and compare your findings.
264
6
Generalized Linear Mixed Models for Proportions and Percentages
Table 6.55 Greenhouse experiment results of cucumber varieties
Block
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
Section
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
Plot
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
16
16
17
17
18
18
Weather
2
2
2
2
2
2
1
1
1
1
1
1
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
Fertilizer
2
2
3.5
3.5
4
4
2
2
3.5
3.5
4
4
2
2
3.5
3.5
4
4
2
2
3.5
3.5
4
4
2
2
3.5
3.5
4
4
2
2
3.5
3.5
4
4
Variety
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Aminex
Dalibor
Proportion (%)
48.8981
42.2463
48.2108
41.6767
55.4369
40.9562
51.5573
36.7739
47.9937
47.8723
57.9171
37.7185
60.1747
45.6937
51.0017
52.2796
51.1251
48.7217
51.6001
50.4463
48.3387
38.6538
51.3147
38.2488
49.6958
29.6786
46.6692
36.5892
56.032
36.0955
45.979
37.2489
40.7277
38.4831
44.5242
34.3907
Level
0.06915
0.06595
0.04679
0.04881
0.04025
0.04859
0.09353
0.10353
0.05327
0.04397
0.05225
0.09324
0.04182
0.06983
0.08863
0.03622
0.05875
0.08169
0.07001
0.09907
0.05788
0.06834
0.05695
0.07908
0.07218
0.11351
0.08825
0.09107
0.04532
0.08712
0.08882
0.12796
0.06418
0.0854
0.06215
0.09651
Appendix
265
Table 6.56 Rodent experiment results
Dose
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y
0
0
1
1
1
2
0
0
1
2
0
0
3
1
0
0
2
3
0
2
0
0
2
1
1
0
0
N
15
3
9
12
13
13
16
11
11
8
14
13
14
13
8
13
14
14
11
12
15
15
14
11
16
12
14
Dose
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
Y
0
1
1
0
2
0
0
3
0
2
3
1
1
0
0
0
1
0
2
2
2
3
1
0
1
1
1
N
6
14
12
10
14
12
14
14
10
12
13
11
11
11
13
10
12
11
10
12
15
12
12
12
12
13
15
Y
1
0
0
0
0
0
4
0
0
1
2
0
1
1
0
0
1
0
1
0
0
1
2
1
0
0
1
Dose
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
N
12
12
11
13
12
14
15
14
12
6
13
10
14
12
10
9
12
13
14
13
14
13
12
14
13
12
7
Dose
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
Y
12
1
0
2
2
4
0
1
0
1
3
0
1
0
3
2
3
3
1
1
8
0
2
8
4
2
N
12
12
13
8
12
13
13
13
12
9
9
11
14
10
12
21
10
11
11
11
14
15
13
11
12
12
Appendix
Data: Fleas
Bioen
B1
B1
B1
B1
B1
B1
B1
B1
B1
SP
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Treat
T1
T1
T1
T2
T2
T2
T3
T3
T3
Rep
1
2
3
1
2
3
1
2
3
Overvi
10
10
10
10
10
10
9
9
8
Dead
0
0
0
0
0
0
1
1
2
(continued)
266
Data: Fleas
Bioen
B1
B1
B1
B1
B1
B1
B1
B1
B1
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
6
SP
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Daphnia
Generalized Linear Mixed Models for Proportions and Percentages
Treat
T4
T4
T4
T5
T5
T5
T6
T6
T6
T1
T1
T1
T2
T2
T2
T3
T3
T3
T4
T4
T4
T5
T5
T5
T6
T6
T6
T1
T1
T1
T2
T2
T2
T3
T3
T3
T4
T4
T4
T5
T5
T5
Rep
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
Overvi
2
2
3
0
0
0
0
0
0
10
10
10
10
10
10
9
9
9
2
2
2
0
0
0
0
0
0
10
10
10
10
10
10
8
9
9
3
2
2
0
0
0
Dead
8
8
7
10
10
10
10
10
10
0
0
0
0
0
0
1
1
1
8
8
8
10
10
10
10
10
10
0
0
0
0
0
0
2
1
1
7
8
8
10
10
10
(continued)
Appendix
Data: Fleas
Bioen
B3
B3
B3
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B3
B3
B3
267
SP
Daphnia
Daphnia
Daphnia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Treat
T6
T6
T6
T1
T1
T1
T2
T2
T2
T3
T3
T3
T4
T4
T4
T5
T5
T5
T6
T6
T6
T1
T1
T1
T2
T2
T2
T3
T3
T3
T4
T4
T4
T5
T5
T5
T6
T6
T6
T1
T1
T1
Rep
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
Overvi
0
0
0
10
10
10
5
6
6
5
5
5
2
3
3
2
2
2
0
0
0
10
10
10
7
5
6
5
5
5
4
4
4
2
2
2
0
0
0
10
10
10
Dead
10
10
10
0
0
0
5
4
4
5
5
5
8
7
7
8
8
8
10
10
10
0
0
0
3
5
4
5
5
5
6
6
6
8
8
8
10
10
10
0
0
0
(continued)
268
Data: Fleas
Bioen
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
B3
6
SP
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Dubia
Generalized Linear Mixed Models for Proportions and Percentages
Treat
T2
T2
T2
T3
T3
T3
T4
T4
T4
T5
T5
T5
T6
T6
T6
Data: Commercial crop explant detachment
Block
A
B
1
1
1
2
1
1
1
1
1
2
1
1
1
1
2
2
1
2
1
1
2
2
1
2
1
2
1
2
2
1
1
2
1
2
2
1
1
2
2
2
2
2
1
2
2
2
2
2
1
3
1
2
3
1
1
3
1
2
3
1
1
3
2
2
3
2
1
3
2
2
3
2
Rep
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
Overvi
8
8
7
5
5
6
2
3
2
3
2
2
0
0
0
Dead
2
2
3
5
5
4
8
7
8
7
8
8
10
10
10
C
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
y
15
10
17
19
26
21
14
12
10
12
30
32
37
30
32
37
18
18
23
21
24
27
37
37
N
73
86
69
32
125
62
81
21
92
108
44
33
91
42
98
44
52
73
108
55
106
92
64
97
Appendix
Data: Cockroaches (E1 = np, E2 = ng, E3 = adult)
Bioassay
Isolation
1
Bb1
2
Bb1
1
Bb1
2
Bb1
1
Bb1
2
Bb1
1
Bb2
2
Bb2
1
Bb2
2
Bb2
1
Bb2
2
Bb2
1
Bb3
2
Bb3
1
Bb3
2
Bb3
1
Bb3
2
Bb3
1
Bb4
2
Bb4
1
Bb4
2
Bb4
1
Bb4
2
Bb4
1
Bb5
2
Bb5
1
Bb5
2
Bb5
1
Bb5
2
Bb5
1
Bb6
2
Bb6
1
Bb6
2
Bb6
1
Bb6
2
Bb6
1
Bb8
2
Bb8
1
Bb8
2
Bb8
1
Bb8
2
Bb8
269
Age
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
Dead
7
6
2
2
9
6
6
7
7
3
10
8
3
2
2
3
8
9
6
4
5
3
10
8
7
7
3
1
7
3
7
9
6
2
10
7
9
9
4
2
9
10
(continued)
270
6 Generalized Linear Mixed Models for Proportions and Percentages
Data: Cockroaches (E1 = np, E2 = ng, E3 = adult)
Bioassay
Isolation
1
Bb9
2
Bb9
1
Bb9
2
Bb9
1
Bb9
2
Bb9
1
Bb10
2
Bb10
1
Bb10
2
Bb10
1
Bb10
2
Bb10
1
Bb11
2
Bb11
1
Bb11
2
Bb11
1
Bb11
2
Bb11
1
Bb12
2
Bb12
1
Bb12
2
Bb12
1
Bb12
2
Bb12
1
Bb13
2
Bb13
1
Bb13
2
Bb13
1
Bb13
2
Bb13
1
Bb14
2
Bb14
1
Bb14
2
Bb14
1
Bb14
2
Bb14
1
Bb15
2
Bb15
1
Bb15
2
Bb15
1
Bb15
2
Bb15
Age
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
np
np
ng
ng
a
a
Dead
5
8
6
2
7
5
8
6
1
4
3
4
8
7
1
3
6
8
8
9
8
9
7
6
6
3
0
1
5
6
10
5
4
2
6
6
5
10
6
1
4
5
(continued)
Appendix
Data: Cockroaches (E1 = np, E2 = ng, E3 = adult)
Bioassay
Isolation
1
Bb16
2
Bb16
1
Bb16
2
Bb16
1
Bb16
2
Bb16
1
Control
2
Control
1
Control
2
Control
1
Control
2
Control
271
Age
np
np
ng
ng
a
a
np
np
ng
ng
a
a
Dead
5
7
3
4
8
6
1
0
0
0
0
1
Data: Disease incidence in grapevine plants (b = block, v = plant, r = shoot, t = treatment,
m = number of diseased leaves per shoot, and n = total number of leaves per shoot).
b
v
r
t
M
n
1
1
1
1
1
14
1
1
1
2
2
12
1
1
1
3
0
12
1
1
1
4
0
13
1
1
1
5
3
8
1
1
1
6
0
9
1
1
2
1
7
8
1
1
2
2
0
10
1
1
2
3
1
14
1
1
2
4
0
10
1
1
2
5
0
17
1
1
2
6
0
10
1
1
3
1
9
14
1
1
3
2
1
11
1
1
3
3
0
10
1
1
3
4
1
14
1
1
3
5
0
10
1
1
3
6
0
21
1
1
4
1
10
17
1
1
4
2
0
9
1
1
4
3
1
12
1
1
4
4
0
11
1
1
4
5
0
12
1
1
4
6
0
10
1
1
5
1
8
11
1
1
5
2
1
10
(continued)
272
6 Generalized Linear Mixed Models for Proportions and Percentages
Data: Disease incidence in grapevine plants (b = block, v = plant, r = shoot, t = treatment,
m = number of diseased leaves per shoot, and n = total number of leaves per shoot).
b
v
r
t
M
n
1
1
5
3
0
9
1
1
5
4
2
12
1
1
5
5
0
10
1
1
5
6
1
11
1
2
1
1
7
9
1
2
1
2
2
10
1
2
1
3
0
10
1
2
1
4
0
14
1
2
1
5
1
12
1
2
1
6
0
13
1
2
2
1
6
12
1
2
2
2
0
11
1
2
2
3
1
13
1
2
2
4
0
9
1
2
2
5
2
11
1
2
2
6
0
10
1
2
3
1
6
7
1
2
3
2
1
12
1
2
3
3
0
9
1
2
3
4
1
10
1
2
3
5
0
14
1
2
3
6
2
12
1
2
4
1
7
13
1
2
4
2
0
10
1
2
4
3
0
10
1
2
4
4
1
12
1
2
4
5
0
9
1
2
4
6
1
8
1
2
5
1
11
15
1
2
5
2
1
13
1
2
5
3
0
14
1
2
5
4
1
14
1
2
5
5
0
11
1
2
5
6
0
11
1
3
1
1
5
11
1
3
1
2
5
11
1
3
1
3
0
15
1
3
1
4
1
15
1
3
1
5
0
8
1
3
1
6
1
10
1
3
2
1
4
9
1
3
2
2
1
15
(continued)
Appendix
273
Data: Disease incidence in grapevine plants (b = block, v = plant, r = shoot, t = treatment,
m = number of diseased leaves per shoot, and n = total number of leaves per shoot).
b
v
r
t
M
n
1
3
2
3
0
11
1
3
2
4
0
13
1
3
2
5
1
12
1
3
2
6
0
12
1
3
3
1
9
12
1
3
3
2
2
14
1
3
3
3
0
12
1
3
3
4
0
12
1
3
3
5
0
10
1
3
3
6
0
13
1
3
4
1
10
10
1
3
4
2
0
10
1
3
4
3
0
8
1
3
4
4
0
10
1
3
4
5
1
14
1
3
4
6
3
11
1
3
5
1
9
11
1
3
5
2
0
11
1
3
5
3
1
11
1
3
5
4
1
14
1
3
5
5
0
9
1
3
5
6
0
9
2
1
1
1
0
12
2
1
1
2
0
12
2
1
1
3
0
14
2
1
1
4
0
12
2
1
1
5
0
10
2
1
1
6
1
13
2
1
2
1
8
9
2
1
2
2
1
9
2
1
2
3
0
12
2
1
2
4
0
10
2
1
2
5
0
12
2
1
2
6
1
10
2
1
3
1
11
14
2
1
3
2
1
12
2
1
3
3
1
11
2
1
3
4
0
10
2
1
3
5
0
9
2
1
3
6
3
11
2
1
4
1
12
15
2
1
4
2
0
15
(continued)
274
6 Generalized Linear Mixed Models for Proportions and Percentages
Data: Disease incidence in grapevine plants (b = block, v = plant, r = shoot, t = treatment,
m = number of diseased leaves per shoot, and n = total number of leaves per shoot).
b
v
r
t
M
n
2
1
4
3
0
10
2
1
4
4
1
9
2
1
4
5
1
10
2
1
4
6
0
16
2
1
5
1
10
14
2
1
5
2
1
9
2
1
5
3
0
11
2
1
5
4
0
11
2
1
5
5
0
11
2
1
5
6
0
11
2
2
1
1
1
9
2
2
1
2
0
9
2
2
1
3
0
12
2
2
1
4
1
10
2
2
1
5
1
12
2
2
1
6
0
17
2
2
2
1
9
12
2
2
2
2
0
12
2
2
2
3
0
11
2
2
2
4
2
14
2
2
2
5
0
11
2
2
2
6
0
10
2
2
3
1
7
13
2
2
3
2
0
16
2
2
3
3
1
12
2
2
3
4
0
10
2
2
3
5
0
10
2
2
3
6
0
11
2
2
4
1
7
13
2
2
4
2
1
18
2
2
4
3
0
10
2
2
4
4
0
11
2
2
4
5
0
11
2
2
4
6
3
13
2
2
5
1
5
10
2
2
5
2
0
10
2
2
5
3
0
10
2
2
5
4
0
10
2
2
5
5
0
9
2
2
5
6
1
12
2
3
1
1
6
13
2
3
1
2
0
10
(continued)
Appendix
275
Data: Disease incidence in grapevine plants (b = block, v = plant, r = shoot, t = treatment,
m = number of diseased leaves per shoot, and n = total number of leaves per shoot).
b
v
r
t
M
n
2
3
1
3
1
11
2
3
1
4
3
11
2
3
1
5
0
12
2
3
1
6
1
19
2
3
2
1
12
13
2
3
2
2
0
11
2
3
2
3
0
8
2
3
2
4
0
9
2
3
2
5
0
17
2
3
2
6
0
12
2
3
3
1
8
11
2
3
3
2
4
12
2
3
3
3
0
11
2
3
3
4
0
10
2
3
3
5
0
15
2
3
3
6
3
13
2
3
4
1
5
14
2
3
4
2
1
9
2
3
4
3
0
12
2
3
4
4
1
12
2
3
4
5
0
10
2
3
4
6
2
14
2
3
5
1
10
14
2
3
5
2
0
14
2
3
5
3
1
10
2
3
5
4
1
13
2
3
5
5
1
15
2
3
5
6
4
10
3
1
1
1
8
12
3
1
1
2
1
14
3
1
1
3
0
12
3
1
1
4
0
20
3
1
1
5
1
18
3
1
1
6
7
15
3
1
2
1
9
16
3
1
2
2
1
12
3
1
2
3
0
13
3
1
2
4
0
15
3
1
2
5
0
17
3
1
2
6
1
18
3
1
3
1
7
12
3
1
3
2
0
14
(continued)
276
6 Generalized Linear Mixed Models for Proportions and Percentages
Data: Disease incidence in grapevine plants (b = block, v = plant, r = shoot, t = treatment,
m = number of diseased leaves per shoot, and n = total number of leaves per shoot).
b
v
r
t
M
n
3
1
3
3
1
13
3
1
3
4
0
18
3
1
3
5
0
14
3
1
3
6
0
14
3
1
4
1
10
14
3
1
4
2
2
17
3
1
4
3
0
10
3
1
4
4
1
19
3
1
4
5
0
17
3
1
4
6
0
16
3
1
5
1
9
10
3
1
5
2
1
14
3
1
5
3
1
11
3
1
5
4
0
18
3
1
5
5
0
15
3
1
5
6
1
11
3
2
1
1
10
10
3
2
1
2
1
11
3
2
1
3
0
12
3
2
1
4
1
15
3
2
1
5
4
20
3
2
1
6
0
14
3
2
2
1
9
12
3
2
2
2
1
10
3
2
2
3
1
12
3
2
2
4
3
18
3
2
2
5
0
16
3
2
2
6
0
12
3
2
3
1
10
11
3
2
3
2
1
16
3
2
3
3
1
14
3
2
3
4
1
17
3
2
3
5
2
15
3
2
3
6
1
16
3
2
4
1
9
11
3
2
4
2
2
14
3
2
4
3
0
10
3
2
4
4
0
18
3
2
4
5
0
17
3
2
4
6
0
12
3
2
5
1
11
12
3
2
5
2
2
12
(continued)
Appendix
Data: Disease incidence in grapevine plants (b = block, v = plant, r = shoot, t = treatment,
m = number of diseased leaves per shoot, and n = total number of leaves per shoot).
b
v
r
t
M
n
3
2
5
3
0
11
3
2
5
4
0
13
3
2
5
5
0
18
3
2
5
6
0
12
3
3
1
1
7
9
3
3
1
2
0
13
3
3
1
3
0
9
3
3
1
4
0
18
3
3
1
5
0
18
3
3
1
6
0
13
3
3
2
1
6
14
3
3
2
2
3
16
3
3
2
3
1
15
3
3
2
4
0
17
3
3
2
5
1
17
3
3
2
6
3
14
3
3
3
1
10
11
3
3
3
2
0
10
3
3
3
3
1
16
3
3
3
4
1
18
3
3
3
5
0
16
3
3
3
6
0
11
3
3
4
1
10
10
3
3
4
2
1
14
3
3
4
3
0
10
3
3
4
4
1
19
3
3
4
5
2
19
3
3
4
6
2
14
3
3
5
1
8
10
3
3
5
2
0
12
3
3
5
3
0
12
3
3
5
4
0
18
3
3
5
5
0
14
3
3
5
6
0
12
277
278
6 Generalized Linear Mixed Models for Proportions and Percentages
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Chapter 7
Time of Occurrence of an Event of Interest
7.1
Introduction
In studies such as biological sciences, animal science, and agronomy, a common
outcome of interest is the time at which an event of interest occurs. The main
characteristic of these data is that the subjects/experimental units are usually
observed for different periods of time until the event of interest occurs. These events
of interest may be adverse events such as the death of an experimental unit and the
cessation of lactation, or positive events such as the conception of a female’s
offspring from a particular treatment and the onset of estrus in a female undergoing
hormone treatment, among others. Because of the characteristics of these response
variables, a “normal” distribution is often a poor choice for modeling the time at
which the event of interest occurs. Exponential, log-normal, gamma, Weibull, and
other more complex distributions that tend to be more common and are better
choices for modeling these phenomena.
Fitting a generalized linear mixed model (GLMM) is a good option for analyzing
these phenomena because the conditional response distribution of the random effects
of this model has desirable properties. In this vein, it is conventional to speak of
survival data and survival analysis, regardless of the nature of the event. Similar data
also arise when measuring the time to complete a task, such as walking 50 meters,
passing an agronomy exam, performing a sensory evaluation of coffee, and so
on. The purpose of this chapter is to provide the reader with the essential language
of linear models and the connection between GLMMs and survival analysis.
© The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8_7
279
280
7.2
7
Time of Occurrence of an Event of Interest
Generalized Linear Mixed Models with a Gamma
Response
The gamma family of distributions encompasses continuous, nonnegative, right-skewed
values. A gamma distribution has two nonnegative parameters –α and β –the probability
density function of which is given by:
f ðy; α, βÞ =
1
-y
yα - 1 ef =βg , y ≥ 0
Γ ðαÞβα
1
where Γ ðαÞ = 0 t α - 1 e - t dt is the gamma function (Casella and Berger 2002). The
mean and variance of a random gamma variable are E[Y] = αβ = μ and
Var½Y = αβ2 = μ2=α, respectively. This density function can be rewritten in terms
of the mean μ and the scale parameter ϕ = 1/α.
f ðy; α, βÞ =
7.2.1
1
Γ
1
ϕ
1=ϕ 1 - 1
ϕ
ðμϕÞ y
ef
- y=μϕg
, y ≥ 0:
CRD: Estrus Induction in Pelibuey Ewes
Estrus induction in ewes is a very common practice carried out in livestock farms or
at research centers. For this, an animal researcher uses gonadotropin-releasing
hormone (GnRH), equine chorionic gonadotropin (eCG), and P4 in a controlled
internal drug-releasing (CIDR) intravaginal device in female Pelibuey ewes (n = 78)
with single, double, and triple lambing as treatments. In order to ensure that all
animals were in good condition during the experiment, ewes received the same
zootechnical management and feeding. For this experiment, the ewes were synchronized on the same day under a synchronization protocol. Table 7.1 presents the
analysis of variance (ANOVA).
The variables evaluated in this experiment were the time of onset and duration of
estrus (yij) in hours according to the type of calving. The variability among
female sheep on weight, age, and body condition must be taken into account in the
Table 7.1 Sources of
variation and degrees
of freedom
Sources of variation
Treatment
Error
Degrees of freedom
t-1=3-1=2
3
i=
Total
3
i=
r i - t = 75
r i - 1 = 78 - 1 = 77
7.2 Generalized Linear Mixed Models with a Gamma Response
281
Table 7.2 Results of the analysis of variance
(a) Covariance parameter estimates
Start of estrus
Estimate
Standard error
-0.01572
.
Cov Parm
Parto (animal) ð^
σ2birthtypeðanimalÞ Þ
0.06668
Residual ϕ
0.01232
Duration of estrus
Estimate
Standard error
0.08370
0.09692
0.2073
0.08938
(b) Type III tests of fixed effects
Effect
Birth type
Num DF
2
Den DF
75
Inicio estro
F-value
5.12
Pr > F
0.0082
Duración estro
F-value
Pr > F
22.61
<0.0001
analysis. The data from this experiment can be found in the Appendix 1 of this book
(Data: Pelibuey Sheep). Thus, the components of a gamma GLMM are as follows:
Distributions: yij j τðrÞij Gamma μij , ϕ ; i = 1, 2, 3; j = 1, ⋯, r i :
rðτÞij N 0, σ 2τðanimalÞ
Linear predictor: ηij = μ þ τi þ τðr Þij
Link function: log μij = ηij
where ηij is the ith link function for treatment i (type of birth angle, double or triple)
in ewes j, μ is the overall mean, τi is the fixed effect due to type of birth (treatment),
r(τ)ij is the random effect due to type of birth (treatment) in ewes j with
τðr Þj N 0, σ 2τðanimalÞ .
The following GLIMMIX program fits the model
proc glimmix nobound method=laplace;
class animal birthtype;
model Inestro = birthtype/dist=gamma;
random birthtype (animal);
lsmeans birthtype/lines ilink;
run;
Part of the results is reported in Table 7.2.
Subsection (a) shows the estimated variance components due to the type of
parturition used in females σ 2birthtypeðanimalÞ = - 0:0157ð ± 0:0837Þ as well as the
scale parameter ϕ = 0:06668 .
Table 7.2 (b) shows the results of the hypothesis tests for type III fixed effects,
which indicate that there is a statistically significant effect of treatment (type of birth)
on the time of onset and duration of ewe estrus.
282
7
Time of Occurrence of an Event of Interest
Table 7.3 Means and standard errors on the model scale (“Estimate” column) and the data scale
(“Mean” column) for the onset and duration of estrus in Pelibuey ewe lambs
Parto least squares means
Standard
Birth_type Estimate error
Start of estrus
1
3.2913
0.06631
2
3.0622
0.04606
3
3.0496
0.04542
Duration of estrus
1
1.6826
0.1518
2
2.6716
0.1171
3
2.8075
0.09846
DF
t-value
Pr > |t|
Mean
Standard error
mean
75
75
75
49.63
66.48
67.14
<0.0001
<0.0001
<0.0001
26.8787
21.3735
21.1059
1.7824
0.9845
0.9586
75
75
75
11.09
22.81
28.51
<0.0001
<0.0001
<0.0001
5.3795
14.4637
16.5684
0.8164
1.6938
1.6313
The last two columns of Table 7.3, labeled “Mean” and “Standard error,”
correspond to the means (μij) on the data scale for the ewes’ mean onset and duration
of estrus with their respective standard errors. For example, the mean time to onset of
estrus in single-birth ewes was 26.87 ± 1.78 hours, whereas for double- and triplebirth ewes, it was 21.37 ± 0.98 and 21.1 ± 0.95, respectively. On the other hand, the
average time (in hours) of estrus duration was longer in double- and triple-birth ewes
(14.46 ± 1.69 and 16.56 ± 1.63, respectively) compared to single-birth ewes
(5.38 ± 0.81).
7.2.2
Randomized Complete Block Design (RCBD): Itch
Relief Drugs
A total of 10 male volunteer patients between 20 and 30 years of age participated as a
study group to compare 7 treatments (Trts) (5 drugs, 1 placebo, and 1 no drug) to
relieve their itching. Since each subject responded differently to each drug, and, in
addition, each subject received a different treatment in the 7 days of study, each of
the subjects can be considered a block. Treatment assignment was randomized
across days. Except for the drug-free day, subjects were administered the treatment
intravenously, and, then, their forearms were induced to itch using an effective itch
stimulus called cowage. The duration of itching, in seconds, was recorded. The data
are shown in Table 7.4.
From left to right, the drugs used were papaverine = Papv, morphine = Morp,
aminophylline = Amino, pentobarbital = Pent, and tripelennamine pentobarbital = Tripel.
The analysis of variance table (Table 7.5) shows the sources of variation and
degrees of freedom for this experiment.
7.2
Generalized Linear Mixed Models with a Gamma Response
283
Table 7.4 Time taken to get rid of the itch
Patient
1
2
3
4
5
6
7
8
9
10
No drug
174
224
260
255
165
237
191
100
115
189
Table 7.5 Sources of
variation and degrees of
freedom
Placebo
263
213
231
291
168
121
137
102
89
433
Papv
105
103
145
103
144
94
35
133
83
237
Morp
199
143
113
225
176
144
87
120
100
173
Sources of variation
Blocks
Treatment
Error
Total
Amino
141
168
78
164
127
114
96
222
165
168
Pent
108
341
159
135
239
136
140
134
185
188
Tripel
141
184
125
227
194
155
121
129
79
317
Degrees of freedom
r - 1 = 10 - 1 = 9
t-1=7-1=6
(t - 1)(r - 1) = 6 × 9 = 54
r × t - 1 = 10 × 7 - 1 = 69
The components of the GLMM with a gamma response are as follows:
Distributions : yij j rðαβÞijk Gammaðμij , ϕÞ; i = 1, ⋯, 7; j = 1, ⋯, 10:
r j N 0, σ2patient
Linear predictor: ηij = μ þ r j þ τi
Link function: log μijk = ηijk
where ηij is the predictor with treatment i and block j, μ is the overall mean, rj is the
random effect of the patient with r j N 0, σ 2patient , and τi is the fixed effect due to
treatment.
Note, although the exponential and gamma distributions have a canonical link
equal to the inverse of the mean, the gamma and exponential GLMMs most often use
a computationally more stable link (link = log), which was used in this and in the
previous analysis.
The following GLIMMIX syntax adjusts a GLMM into complete blocks.
proc glimmix nobound method=laplace;
class Patient Trt;
model y = Trt/dist=gamma;
random Patient;
lsmeans Trt/lines ilink;
run;
284
7
Table 7.6 Results of the
analysis of variance
Time of Occurrence of an Event of Interest
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Estimate
Patient
0.03964
0.09132
Residual ϕ
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
Trt
6
54
728.62
5.69
0.08
Standard error
0.02375
0.01640
F-value
3.82
Pr > F
0.0030
The statistics of the conditional model (Pearson′s chi - squre/DF = 0.08) as well
as the variance components (Patient) and the scale parameter ϕ of the model
indicate that the gamma model adequately describes the dataset (Table 7.6 parts
(a) and (b)). The analysis of variance (Table 7.6 part (c)) indicates that there is
a highly significant difference of treatments in the mean time of itch duration
(P = 0.0030).
The dispersion observed in the following plot (top left) of the residuals versus the
linear predictor value suggests that the variance is constant and homogeneous
(Fig. 7.1). The histogram (upper right) shows a nearly symmetrical pattern with
little bias. Furthermore, the residuals versus quantile plot (bottom left) shows no
marked deviations, indicating that the fit is adequate. Finally, the bottom right plot
shows that the average residuals are zero and vary between -0.5 and 0.75.
The “lsmeans” on the data scale, for each of the five treatments, placebo, and the
control treatment, are shown under the “Mean” column with their respective “Standard error” in Table 7.7. Each of the five drugs appear to have a significant effect
compared to the placebo and control. Papaverine (Papv) is the most effective drug.
Both the placebo and control treatment have statistically similar means. The relatively large difference in the placebo group suggests that some patients responded
negatively to the placebo compared to the control, whereas others responded
positively.
Figure 7.2 shows that the drug papaverine significantly reduced the itching time,
followed by the drugs aminophylline and morphine, whereas the efficacies of the
drugs pentobarbital and tripelennamine were highly similar to each other in eliminating itching.
7.2.3
Factorial Design: Insect Survival Time
This experiment consisted of studying the effectiveness of four different types of
insecticides (Insec1, Insec2, Insec3, and Insec4) at three different concentration
levels (low, medium, and high) in the survival time (in hours) of a particular species
7.2
Generalized Linear Mixed Models with a Gamma Response
285
Fig. 7.1 Conditional residuals
Table 7.7 Means and standard errors on the model scale (“Estimate” column) and the data scale
(“Mean” column) for the average duration time of the itch
Trt least squares means
Standard
Trt
Estimate error
Amino
4.9795
0.1149
Morp
4.9797
0.1146
Papv
4.7356
0.1149
Pento
5.1703
0.1149
Placebo
5.2704
0.1151
No drug 5.2542
0.1148
Tripel
5.0802
0.1147
DF
t-value
43.32
43.44
41.20
44.99
45.79
45.76
44.28
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
145.41
145.43
113.93
175.97
194.49
191.36
160.80
Standard error
of mean
16.7129
16.6733
13.0956
20.2211
22.3867
21.9723
18.4487
of beetles (Appendix 1: Data: Beetles). The interaction between both factors (insecticide * dose) yielded a total of 12 combinations (treatments). The objective of this
study was to compare the insecticides, dose, and interaction with beetle survival
time. Due to the intrinsic characteristics of each of the insects, these must be
considered as a source of variation in the experiment, since they respond differently
286
7
Time of Occurrence of an Event of Interest
Average me (seconds)
250
200
150
100
50
0
Amino
Morp
Papv
Pento
Placebo
Sindroga
Tripel
Treatment
Fig. 7.2 Average time taken to eliminate itching
Table 7.8 Sources of variation and degrees of freedom
Sources of variation
Blocks
Insecticide
Dose
Insecticide * dosage
Error
Total
Degrees of freedom
r-1=4-1=3
a-1=4-1=3
b-1=3-1=2
(a - 1)(b - 1) = 3 × 2 = 6
ab(r - 1) = 4 × 3 × 3 = 33
r × a × b - 1 = 4 × 4 × 3 - 1 = 47
to certain stimuli. Assuming that 48 beetles are available, they were randomly
assigned equally to 4 groups (blocks) with 12 treatment combinations. That is,
four beetles were randomly assigned to each treatment.
The sources of variation and degrees of freedom for this experiment are shown in
the following analysis of variance table (Table 7.8).
The components of the gamma-response GLMM are as follows:
Distributions : yijk j r k Gammaðμijk , ϕÞ; i = 1, ⋯, 4; j = 1, 2, 3; k = 1, ⋯, 4:
r k Nð0, σ2block Þ
Linear predictor: ηijk = μ þ rk þ αi þ βj þ ðαβÞij
Link function: log μijk = ηijk
The following GLIMMIX command adjusts a GLMM with a gamma response.
7.2
Generalized Linear Mixed Models with a Gamma Response
Table 7.9 Results of the
analysis of variance
(a) Fit statistics for conditional distribution
-2 Log L (tiempo | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Estimate
block
-0.00173
Residual
0.04155
(c) Type III tests of fixed effects
Effect
Num DF Den DF
Dose
2
33
Insecticide
3
33
Dose*insecticide 6
33
287
121.05
1.91
0.04
Standard error
.
0.008818
F-value
69.61
31.36
2.05
Pr > F
<0.0001
<0.0001
0.0868
proc glimmix nobound method=laplace;
class dose insecticide insect;
model time = dose|insecticide/dist=gamma;
random insect;
lsmeans dose|insecticide/lines ilink;
run;
Part of the Statistical Analysis Software (SAS) output is shown in Table 7.9. The
value of the conditional model’s Pearson′s chi - square/DF = 0.04 indicates that the
gamma distribution adequately models the data. The estimated variance component
for blocks and the scaling parameter given by the “residual” value are shown below
^2 = 0:04155, respectivelyÞ.
(in part (b)) ð^
σ2block = - 0:00173, and σ
The analysis of variance in (c) of Table 7.9 indicates that the insecticides and dose
(P = 0.0001) have different significant effectiveness (toxicity) on beetle survival
time. However, the interaction between both factors is close to significance
(P = 0.0868). The “lsmeans” values on the data scale for dose μi:: (part (a)) and
insecticide μ:j: (part (b)) with their respective standard errors for both factors are
listed under the columns titled “Mean” and “Standard error mean” of Table 7.10,
respectively.
The combination of levels of both factors affected the average survival time of the
beetles (Table 7.11). For insecticides 1 and 3 at a high dose, the survival time was
lower with average times of 2.1 ± 0.209 and 2.35 ± 0.334 hours, respectively. In
general, low values of survival times were observed for insecticides 1 and 3 compared to insecticides 2 and 4.
7.2.4
A Split Plot with a Factorial Structure on a Large Plot
in a Completely Randomized Design (CRD)
Four samples were obtained from each of two batches (Reps) of unprocessed gum
from Acacia sp. Trees, with eight samples in total. Within each batch, the four
288
7
Time of Occurrence of an Event of Interest
Table 7.10 Means and standard errors on the model scale (“Estimate”) and the data scale (“Mean”)
for the factor dose and type of insecticide
(a) Dose least squares means
Standard
Dose
Estimate error
High
0.9960
0.04984
Low
1.7840
0.04984
Medium 1.6203
0.04984
(b) Insecticide least squares means
Standard
Insecticide Estimate error
Insec1
1.1074
0.05755
Insec2
1.8272
0.05755
Insec3
1.3041
0.05755
Insec4
1.6284
0.05755
DF
33
33
33
DF
33
33
33
33
t-value
19.98
35.79
32.51
t-value
19.24
31.75
22.66
28.29
Pr > |t|
<0.0001
<0.0001
<0.0001
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
Mean
2.7075
5.9538
5.0548
Mean
3.0265
6.2166
3.6845
5.0960
Standard error
mean
0.1349
0.2967
0.2519
Standard error
mean
0.1742
0.3578
0.2121
0.2933
Table 7.11 Means and standard errors on the model scale and the data scale for the interaction
between dose and type of insecticide
Dose*insecticide least squares means
Standard
Dose
Insecticide Estimate error
High
Insec1
0.7419
0.09968
High
Insec2
1.2089
0.09968
High
Insec3
0.8545
0.09969
High
Insec4
1.1788
0.09969
Low
Insec1
1.4171
0.09968
Low
Insec2
2.1747
0.09968
Low
Insec3
1.7361
0.09969
Low
Insec4
1.8082
0.09968
Medium Insec1
1.1632
0.09969
Medium Insec2
2.0980
0.09968
Medium Insec3
1.3218
0.09969
Medium Insec4
1.8984
0.09969
DF
33
33
33
33
33
33
33
33
33
33
33
33
t-value
7.44
12.13
8.57
11.82
14.22
21.82
17.42
18.14
11.67
21.05
13.26
19.04
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
2.1000
3.3499
2.3501
3.2503
4.1250
8.7998
5.6754
6.0994
3.2000
8.1499
3.7501
6.6753
Standard
error mean
0.2093
0.3339
0.2343
0.3240
0.4112
0.8772
0.5658
0.6080
0.3190
0.8124
0.3738
0.6654
samples were randomly assigned to combinations of two factors with two levels
each. The first factor refers to whether the gum was demineralized or not, and the
second factor refers to whether the gum was pasteurized or not. An emulsion made
from each gum sample was divided into three smaller parts, which were randomly
assigned to the levels of a third factor, the PH, and pH was adjusted to 2.5, 4.5, or 5.5
using citric acid (Appendix 1: Data: Gum Breakdown Times).
This is a split-plot design, with whole plots and rubber samples in a block
arrangement. The combined levels of demineralization and pasteurization of the
paste are large (whole) plot factors. The split plots are the smaller parts, with a
specific pH, which is the only split-plot factor. The response measured ( y) was the
7.2
Generalized Linear Mixed Models with a Gamma Response
289
Table 7.12 Sources of variation and degrees of freedom
Sources of variation
Demineralization (Des)
Pasteurization (Pasteu)
Demineralization*pasteurization
Des*Pasteu (rep)
pH
Demineralization*pH
Pasteurization*pH
Des*Pasteu*pH
Error
Total
Degrees of freedom
a-1=2-1=1
b-1=2-1=1
(a - 1)(b - 1) = 1
ab(r - 1) = 2 × 2 × 1 = 4
(c - 1) = 3 - 1 = 2
(a - 1)(c - 1) = 2
(b - 1)(c - 1) = 2
(a - 1)(b - 1)(c - 1) = 2
ab(c - 1)(r - 1) = 2 × 2 × 2 × 1 = 8
r × a × b × c - 1 = 2 × 2 × 2 × 3 - 1 = 23
time to break, i.e., the time (in hours) until the emulsion failed. The sources of
variation and degrees of freedom for this experiment are shown in Table 7.12.
The components of the GLMM with a Gamma response are as follows:
Distributions: yijkl j r l , αβðr Þijl Gamma μijkl , ϕ ; i = 1, 2; j = 1, 2; k = 1, 2, 3; l = 1, 2:
r l N 0, σ 2r , αβðr Þijl N 0, σ 2rαβ
Linear predictor: ηijkl = μ þ αi þ βj þ ðαβÞij þ r ðαβÞijl þ γ k þ ðαγ Þik þ ðβγ Þjk
þ ðαβγ Þijk ;
where αi, βj, and γ k are the fixed effects due to the factors demineralization,
pasteurization, and pH, respectively; the effects (αβ)ij, (αγ)ik, (βγ)jk, and (αβγ)ijk
are the two- and three-way interactions of the factors under study; and αβ(r)ijl are
random effects due to the demineralization x pasteurization x rep interaction,
2
.
assuming that αβðr Þijl N 0, σ rαβ
Link function: log μijk = ηijk
The GLIMMIX commands for setting this GLMM are as follows:
proc glimmix nobound method=laplace;
class Batch Demineralization Pasteurization pH;
model y = Demineralization|Pasteurization|pH/dist=gamma;
random batch(Demineralization*Pasteurization);
lsmeans Demineralization|Pasteurization|pH/lines ilink;
run;
290
7
Time of Occurrence of an Event of Interest
Table 7.13 Results of the analysis of variance
(a) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Rep (Desmin*Pasteur)
Residual ϕ
(c) Type III tests of fixed effects
Effect
Demineralization (Des)
Pasteurization (Pasteu)
Demineralization*pasteurization
pH
Demineralization*pH
Pasteurization*pH
Des*Pasteu*pH
192.24
0.12
0.01
Estimate
0.001428
0.006011
Num DF
1
1
1
2
2
2
2
Standard error
0.001864
0.002126
Den DF
4
4
4
8
8
8
8
F-value
35.48
19.49
35.67
5.27
3.84
0.57
4.32
Pr > F
0.0040
0.0116
0.0039
0.0346
0.0676
0.5889
0.0535
The relevant results from the SAS output are shown in Table 7.13. The value of
χ2
the conditional model DF
= 0:01 indicates that the gamma distribution does not
cause overdispersion. The variance component due to blocks × demineralization ×
pasteurization σ 2rðαβÞ and the scale parameter ϕ are shown in (b).
The hypothesis tests for type III fixed effects are presented in part (c) of
Table 7.13, where a significant effect of the factors demineralization, pasteurization,
and pH as well as the interaction between demineralization with pasteurization are
observed on the gum. However, the interactions demineralization*pH (P = 0.0676)
and demineralization*pasteurization*pH are close to significance (P = 0.0535). The
emulsion breaking time is strongly affected by no demineralization (demineralization = 1) and no pasteurization (pasteurization = 1) of the gum and, to a lesser
extent, by the pH adjusted to the gum (Table 7.14).
Analyzing the simple effects of the factors, we can observe that when the gum has
not been pasteurized (B = 1), the average emulsion break time is very similar in the
demineralized paste than in the non-demineralized paste at the three pH levels.
However, when the gum has been pasteurized, demineralization has a significant
impact on the emulsion breakup time; for example, for a paste that is not
demineralized and pasteurized (A1B2), the emulsion breakup time is much lower
than when the gum has been demineralized and pasteurized (A2B2) at all three pH
levels. Finally, with a demineralized, pasteurized gum at pH = 4.5, a gum with
higher breaking stability is obtained (Table 7.15).
7.3
Survival Analysis
291
Table 7.14 Means and standard errors of the main effects on the model scale (Estimate) and the
data scale (Mean)
(a) Demineralization least squares means
Standard
Demineralization Estimate error
1
5.0911
0.02930
2
5.3379
0.02930
(b) Pasteurization least squares means
Standard
Pasteurization Estimate error
1
5.1230
0.02930
2
5.3059
0.02930
(c) pH least squares means
pH Estimate Standard error DF
1
5.1610
0.03050
8
2
5.2839
0.03050
8
3
5.1986
0.03052
8
DF
4
4
DF
4
4
t-value
173.77
182.18
t-value
174.87
181.08
t-value
169.22
173.24
170.32
Pr > |t|
<0.0001
<0.0001
Pr > |t|
<0.0001
<0.0001
Pr > |t|
<0.0001
<0.0001
<0.0001
Mean
162.57
208.07
Mean
167.84
201.53
Mean
174.33
197.13
181.02
Standard error
mean
4.7628
6.0964
Standard error
mean
4.9171
5.9051
Standard error mean
5.3171
6.0124
5.5255
Table 7.15 Means and standard errors of the simple effects on the model scale (Estimate) and the
data scale (Mean)
Demineralization*pasteurization*pH least squares means
Standard
A B C Estimate error
DF t-value Pr > |t|
1 1 1 5.0696
0.06099
8
83.13
<0.0001
1 1 2 5.1695
0.06105
8
84.68
<0.0001
1 1 3 5.1311
0.06100
8
84.12
<0.0001
1 2 1 5.1137
0.06099
8
83.84
<0.0001
1 2 2 5.0445
0.06098
8
82.72
<0.0001
<0.0001
1 2 3 5.0183
0.06098
8
82.29
2 1 1 5.0811
0.06103
8
83.26
<0.0001
2 1 2 5.1694
0.06099
8
84.76
<0.0001
2 1 3 5.1175
0.06110
8
83.76
<0.0001
2 2 1 5.3796
0.06100
8
88.19
<0.0001
2 2 2 5.7520
0.06100
8
94.30
<0.0001
2 2 3 5.5277
0.06106
8
90.53
<0.0001
Mean
159.11
175.83
169.20
166.28
155.17
151.15
160.95
175.81
166.91
216.93
314.81
251.57
Standard error
mean
9.7035
10.7339
10.3204
10.1419
9.4623
9.2170
9.8225
10.7225
10.1978
13.2320
19.2031
15.3607
A = demineralization (1 = no, 2 = yes), B = pasteurization (1 = no, 2 = yes), and C = pH (1 = 2.5,
2 = 4.5, and 3 = 5.5)
7.3
Survival Analysis
When a research focuses on the time of occurrence of a specific event, we usually
refer to survival times, and, hence, the statistical analysis of these times, as mentioned above, is known as survival analysis. A very characteristic feature of survival
292
7 Time of Occurrence of an Event of Interest
times is the presence of censored times, that is, when there are individuals whose
actual survival time is not known.
For a set of survival times (including censored ones) of a sample of individuals, it
is possible to estimate the proportion of the population that will survive a time
interval under the same circumstances. The methods used to make this estimate are
based on the proposal of Kaplan and Meier (1958). This method allows – through
different statistical tests (log rank, Breslow, Tarone–Ware, etc.) – the comparison of
the survival of two or more groups of individuals who differ with respect to certain
factors.
Survival analysis focuses its interest on a group or several groups of individuals
for whom an event is defined, which occurs after a time interval. To determine the
time of interest, there are three requirements: an initial time, a scale to measure the
passage of time (minutes, hours, days, etc.), and clarity about what is meant by the
event of interest.
Survival of an individual is conceptually the probability of being alive in a given
time "t" from diagnosis, i.e., initiation of treatment or complete remission for a group
of individuals. In clinical studies, survival times often refer to time till death,
development of a particular symptom, or relapse after complete remission of a
disease. Failure is defined as death, relapse, or the occurrence of a new disease. In
many survival analyses, when the end of the observation period previously set by the
investigator is reached, there are individuals to whom the event has not occurred and
we do not know when it will occur. Therefore, the actual survival time for them is
unknown, and only the survival time to the end of the study is known. Such survival
times are called censored times. It also happens, in some cases, that some individuals
do not continue the study until the end of the analysis period for reasons unrelated to
the research, e.g., death from other causes; these times are also censored. These
censored data contribute valuable information and, therefore, should not be omitted
from the analysis.
The pharmaceutical and food industries are legally required to label the shelf life
of their product on the packaging. For pharmaceuticals, the requirements for how to
determine shelf life are highly regulated. However, the regulatory standards do not
specifically define shelf life. Instead, the definition is implicit through the estimation
procedure. The interest is in the situation where multiple batches are used to
determine a shelf life of a product that applies to all future batches. Consequently,
both shelf life and label life are of great importance because of the variability within
and between batches. Product development must be very well thought out before a
company can have confidence in shelf life estimates. The company must be able to
reliably produce a homogeneous product from batch to batch of ingredients, as
physical and chemical factors impact the ability of bacteria to grow, such as pH,
water activity, and uniformity of the mix (moisture distribution, salt, preservative or
food acid) and, consequently, the shelf life of the product. Therefore, products
should be inspected at appropriate times and samples should be tested for critical
stability of physical and chemical characteristics. These tests also provide an opportunity to begin microbiological testing for spoilage organisms. Testing should
continue beyond the intended shelf life unless the product fails earlier. Testing
7.3
Survival Analysis
293
should lead to an understanding of target levels and ranges of ingredients for
evaluation of the critical physical and chemical characteristics of the product over
the intended shelf life.
Survival analysis is the name for a collection of statistical techniques used to
describe and quantify the time in which the event of interest occurs. The term
“survival time” specifies the amount of time taken to occur. Situations in which
survival analyses have been used in epidemiology include:
(a) Survival of insects after having received an insecticide.
(b) The time taken by cows or ewes to conceive after calving.
(c) The time taken for a farm to experience its first case of an exotic disease.
7.3.1
Concepts and Definitions
To clearly understand and interpret a rate of change calculated from the event data of
interest, a more extensive approach is needed. The definition of a rate of change
begins with the mathematical description of a changing pattern over time,
represented by the symbol S(t). A version of a ratio is created by dividing the change
in function S(t)[S(t) to S(t + Δt)] by the corresponding change over time t(t to t + Δt)
producing the rate of change
rate of change =
change on Sðt Þ
Sðt Þ - Sðt þ Δt Þ Sðt Þ - Sðt þ Δt Þ
=
=
change on time
Δt
ðt þ Δt Þ - t
Rates of change, with respect to time, apply to a variety of situations, but one
specific function, traditionally denoted by S(t), is fundamental to the analysis of
survival data. This is called the survival function and is defined as the probability of
surviving (probability of survival) beyond a specific point in time (denoted by t).
That is;
Sðt Þ = Pðsurvival time = 0 at time = t Þ
= Pðsurvival in the interval ½0, t Þ
Equivalent to
Sðt Þ = Pðsurviving beyond time t Þ = PðT ≥ t Þ = 1 - F ðt Þ
where F(t) is the cumulative distribution function with F(t) = P(T ≤ t). Another
important concept in survival analysis is the hazard function h(t). The hazard
function that depends on T is defined as
294
7
hðt Þ = lim
Δt → 0
Time of Occurrence of an Event of Interest
Pðt ≤ T < t þ ΔtjT ≥ t Þ
Δt
such that the following expression can be expressed as
F ðt þ Δt Þ - F ðt Þ
1
×
Δt
PðT ≥ t Þ
hðt Þ = lim
Δt → 0
f ðt Þ
Sð t Þ
hð t Þ =
where f(t) is the probability density function. Any distribution defined by t 2 [0, t)
can serve as a survival distribution. Consequently,
hð t Þ = -
∂
f log Sðt Þg:
t
It then follows that
Sðt Þ = expf - H ðt Þg
where H(t) the cumulative hazard function
t
hðuÞdu
H ðt Þ =
0
Another useful relationship is
H ðt Þ = - log Sðt Þ:
For the simplest model, the exponential model with h(t) = λ (λ is a constant), the
survival function is given by
t
Sðt Þ = exp
-
t
hðuÞdu
λdu = e - λt
= exp -
0
0
with the probability density function given by
f ðt Þ =
∂
Sðt Þ = λe - λt :
t
Thus, the survival function, hazard function, and cumulative risk for the
exponential model is given by:
7.3
Survival Analysis
295
Survival function: Sðt Þ = e - λt
Risk function: hðt Þ =
f ðt Þ λe - λt
= - λt = λ
Sð t Þ
e
t
Cumulative risk function: H ðt Þ =
hðuÞdu =
t
7.3.2
t
λdu = λt:
0
CRD: Aedes aegypti
The objective of this experiment was to test the vulnerability of Aedes aegypti
mosquitoes to different fungal treatments (four treatments). A bioassay was
conducted to determine the survival time of each of the mosquitoes. Three-day-old
mosquitoes were maintained after hatching in 45-cm rearing cages with access to
water but not food. The mosquitoes were kept in rearing cages with water and fed
warm pig blood (37 °C) through a natural membrane (sausage casing) approximately
every 3 days and allowed to oviposit freely during the waiting period. A total of
10 mosquitoes were placed in a chamber to which one of the treatments (four) plus a
control was applied. Here, we present part of the data from a bioassay with four
replicates. The complete data from this trial can be found in the Appendix 1 (Data:
Aedes aegypti).
Treatment
C
C
⋮
C
Mam
Mam
⋮
MaS
MaS
⋮
MaC
MaC
MaC
⋮
Ma1
Ma1
⋮
Ma1
Rep
1
1
⋮
4
1
1
⋮
1
1
⋮
1
1
1
⋮
1
1
⋮
4
Y
8
11
⋮
20
2
2
⋮
3
3
⋮
2
2
2
⋮
2
2
⋮
11
296
7
Table 7.16 Results of the
analysis of variance
Time of Occurrence of an Event of Interest
(a) Fit statistics for conditional distribution
-2 Log L (T | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Trt
4
192
186.42
716.70
35.33
0.18
Pr > F
<0.0001
The components of this GLMM are as follows:
Distributions: yij j repj Gamma μij , ϕ
repj N 0, σ 2rep
Linear predictor: ηij = η þ τi þ repj
Link function: ηij = log μij
where η is the intercept, τi is the treatment effect, and repj is the random effect due to
the mosquito chamber assuming repj N 0, σ 2rep :
The following GLIMMIX commands adjust a GLMM with a gamma response:
proc glimmix data=mosquitos method=laplace;
class bio trt rep;
model y = trt/dist=gamma;
random rep;
lsmeans trt/lines ilink;
run;
Part of the output is shown in Table 7.16. The statistic in (a) above indicates that
there is no over-dispersion in the fit of the data, as indicated by Pearson′s chi square/DF = 0.18. The analysis of variance (type III tests of fixed effects) indicates
that there is a highly significant effect (P = 0.0001) of the fungal treatments on the
mean mosquito survival time.
The relevant information in Table 7.17 “lsmeans” comes from the columns
labeled “Estimate” and “Mean”: these are the estimates on the model scale and the
data scale, and the average survival time in each of the treatments is represented by
μi ð ± standard errorÞ.
The estimated risk function for each treatment combination is ^λi = 1=μ^i : For example, for treatment Ma1, the estimated hazard function is λMa1 = 1=3:4223 = 0:2922: We
can manually calculate these values from the Mean column or we can automate the
process by adding the command “ods output lsmeans = mu” in the GLIMMIX
7.3
Survival Analysis
297
Table 7.17 Means and standard errors of the main effects on the model scale (Estimate) and the
data scale (Mean)
Trt least squares means
Standard
Trt
Estimate error
Ma1
1.2303
0.06354
MaC
0.9562
0.06350
MaS
1.5798
0.06357
Mam
0.6946
0.06350
Control 2.7155
0.06362
DF
192
192
192
192
192
tvalue
19.36
15.06
24.85
10.94
42.68
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
3.4223
2.6017
4.8542
2.0029
15.1126
Standard error
mean
0.2174
0.1652
0.3086
0.1272
0.9615
program above. Once we have saved the treatment means, we can ask SAS to estimate
the estimated hazard function for the treatments. The commands are as follows:
data hazard;
set mu;
hazard=1/mu;
proc print data=hazard;
run;
The results are listed below in Table 7.18. The hazard column contains the
estimated hazard functions for each treatment hi ðt Þ = λi .
From the values λi , we can calculate the estimated survival function Si ðt Þ = e - λi t
for each of the treatments. Figure 7.3 shows the probability of survival over time
^
obtained with Si ðt Þ = e - λi t of each of the proposed treatments and the control.
Clearly, the treatments MaS, Ma1, MaC, and Mam showed a greater efficacy in
the biological control of these mosquitoes.
7.3.3
RCBD: Aedes aegypti
Similar to the previous example, this experiment consisted of testing the vulnerability of Aedes aegypti mosquitoes to different fungal treatments (four treatments). For
this, two bioassays were conducted to determine the survival time of each of the
mosquitoes. Three-day-old mosquitoes were maintained after hatching in 45-cm
rearing cages with access to water but not food. Mosquitoes were maintained in
rearing cages with water and were fed warm pig blood (37 °C) through a natural
membrane (sausage casing) approximately every 3 days. They were allowed to
freely oviposit during the waiting period. A total of 10 mosquitoes were placed in
a chamber to which one of the treatments (four) plus a control was applied. The data
can be found in the Appendix 1 (Data: Aedes aegypti).
Effect
TRT
TRT
TRT
TRT
TRT
TRT
Ma1
MaC
MaS
Mam
Control
Estimate
1.2303
0.9562
1.5798
0.6946
2.7155
Standard error
0.06354
0.06350
0.06357
0.06350
0.06362
DF
192
192
192
192
192
t-value
19.36
15.06
24.85
10.94
42.68
Probt
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
3.4223
2.6017
4.8542
2.0029
15.1126
Standard error mean
0.2174
0.1652
0.3086
0.1272
0.9615
Table 7.18 Means and standard errors of the main effects on the model scale (Estimate) and the data scale (Mean) and the hazard function λi
Hazard λi
0.29220
0.38437
0.20601
0.49927
0.06617
298
7
Time of Occurrence of an Event of Interest
7.3
Survival Analysis
1
299
Ma1
MaC
MaS
Mam
Control
Survival probability (S(t))
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Time (days)
Fig. 7.3 Estimated survival probability for each treatment
The components of this GLMM are as follows:
Distributions: yijk j bioj , repðbioÞkðjÞ Gamma μijk , ϕ
bioj N 0, σ 2bio , repðbioÞkðjÞ N 0, σ 2repðbioÞ
Linear predictor: ηij = η þ τi þ bioj þ repðbioÞkðjÞ
where η is the intercept, τi is the treatment effect, bioj and rep(bio)k( j ) are the random
effects of the bioassay and the mosquito chamber within the bioassay, respectively,
2
assuming bioj N 0, σ 2bio and repðbioÞkðjÞ N 0, σ rep
ðbioÞ :
Link function: ηij = log μij
The following GLIMMIX program fits a block GLMM with a gamma response.
proc glimmix method=laplace nobound;
class bio trt ind rep;
model y = trt/dist=gamma;
random bio rep(bio);
ods output lsmeans=mu;
lsmeans trt/lines ilink;
run;quit;
The results obtained are shown below. Part of the statistics and variance
components are listed in Table 7.19. In part (a), the value of the statistic of
300
7
Table 7.19 Results of the
analysis of variance
Time of Occurrence of an Event of Interest
(a) Fit statistics for conditional distribution
-2 log L (Y | r. effects)
3303.50
Pearson’s chi-square
202.30
Pearson’s chi-square/DF
0.34
(b) Cov Parm
Estimate
Standard error
BIO
0.1859
0.1936
REP(BIO)
0.02562
0.01673
Residual
0.2822
0.01568
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
TRAT
4
588
115.36
<0.0001
Table 7.20 Means and standard errors of the main effects on the model scale (Estimate) and the
data scale (Mean)
TRT least squares means
Standard
TRT
Estimate error
Ma1
1.6344
0.3140
MaC
1.4903
0.3140
MaS
1.8788
0.3140
Mam
1.8053
0.3143
Control 2.8293
0.3139
DF
588
588
588
588
588
tvalue
5.21
4.75
5.98
5.74
9.01
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
5.1266
4.4386
6.5455
6.0820
16.9329
Standard error
mean
1.6097
1.3939
2.0550
1.9115
5.3153
Table 7.21 Means and standard errors of the main effects on the model scale (Estimate), the data
scale (Mean), and the hazard function λi
TRT
Ma1
MaC
MaS
Mam
Control
Estimate
1.6344
1.4903
1.8788
1.8053
2.8293
Standard
error
0.3140
0.3140
0.3140
0.3143
0.3139
DF
588
588
588
588
588
tvalue
5.21
4.75
5.98
5.74
9.01
Probt
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
5.1266
4.4386
6.5455
6.0820
16.9329
Standard error
mean
1.6097
1.3939
2.0550
1.9115
5.3153
Hazard
λi
0.19506
0.22529
0.15278
0.16442
0.05906
Pearson′s chi - square/DF = 0.34 and in part (b), the estimated variance components due to blocks, within-block replicates, and experimental error are
σ^2bio = 0:1859, σ^2repðbioÞ = 0:02562, and σ^2 = 0:2822, respectively. The type III effect
hypothesis tests (part (c)) indicate that there is a highly significant difference
between treatments on the mean survival time, as indicated by P = 0.0001.
Tables 7.20 and 7.21 show the estimates on the model scale and the data scale,
linear predictors ðηi Þ, means ðμi Þ with their respective standard errors, and the
estimated hazard function. The results indicate that the MaC treatment has a greater
lethal effect than A. aegypti mosquito control.
7.4
Exercises
301
1
Ma1
Survival probability (S(t))
0.9
MaC
MaS
Mam
Control
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Time (days)
Fig. 7.4 Estimated survival probability for each treatment
Figure 7.4 shows the survival times for the different treatments tested. These
curves were obtained with Si ðt Þ = e
7.4
- λi t
.
Exercises
Exercise 7.4.1 The investigation of this experiment focused on studying the times
of animal incapacitation experienced after being exposed to the burning of eight
types of aircraft interior materials (M1–M9) and performances in milligram/gram
combustion of seven gases (CO, HCN, H2S, HCl, HBr, NO2, SO2) (Spurgeon 1978).
The recorded incapacitation time of the animal when exposed to different combustion materials (under the column “Material”) is found under the column “Time in
minutes” and in the third column the value of (1000/Time); these data are shown
below (Table 7.22):
(a)
(b)
(c)
(d)
Write down a statistical model of this experiment.
List all the components of the GLMM in (a).
Write down the null and alternative hypotheses associated with this experiment.
Construct an ANOVA table indicating the sources of variation and degrees of
freedom.
(e) Analyze the time of inability of the animal to be exposed to the gases of the
different types of materials.
(f) Comment on the results obtained.
302
7
Time of Occurrence of an Event of Interest
Table 7.22 Time of incapacity of the animal when exposed to different combustion gases
Material
M1
M1
M1
M1
M1
M1
M1
M1
M1
M1
M1
M1
M1
M2
M2
M2
M2
M2
M2
M2
M2
M2
M3
M3
M3
M3
M3
M3
M3
M3
M3
M4
M4
M4
M4
M4
M4
M4
M4
M4
M4
M4
Time
2.36
2.38
2.61
3.07
3.07
3.19
3.7
3.9
4.18
4.7
4.86
5.58
5.85
3.22
3.89
4.79
5.07
5.22
5.82
6.09
8.36
13.02
4.29
4.8
5.04
5.06
5.25
5.5
5.55
7.55
9.58
1.15
2
2.15
2.22
2.23
2.72
2.93
3.07
3.47
4.18
4.64
1000/Time
423.7
420.2
383.1
325.7
325.7
313.5
270.3
256.4
239.2
212.8
205.8
179.2
170.9
310.6
257.1
208.8
197.2
191.6
171.8
164.2
119.6
76.8
233.1
208.3
198.4
197.6
190.5
181.8
180.2
132.5
104.4
869.6
500
465.1
450.5
448.4
367.6
341.3
325.7
288.2
239.2
215.5
CO
164
174
96
101
142
143
147
156
124
101
142
104
90
159
153
161
159
162
106
124
89
88
129
105
108
120
149
28
83
68
28
88
89
63
112
96
78
348
255
112
144
70
HCN
6.4
7.5
4.7
7.5
6.8
8.2
5.2
4.7
3.2
8.9
4.6
3.4
2.3
16.4
2.9
0.6
0
0
3.2
1.5
0.7
0
6
5.8
7.8
11.6
0
9.1
5
5.5
2.4
62.4
41.7
14.9
37.2
7
33.8
1.9
1.9
19.5
3.8
11.2
H2S
0
0
0
0
0
0
0
0
0
0.9
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.4
0
0
2
0
13.4
0
14.2
0
13.9
0
0
10.7
0
6.2
HCl
0
0
33
0
27.6
0
11.3
12
23.3
5.4
19.4
80
34.4
0
0
0
4.6
22
45.2
0
0
0
4.2
0
7.3
23
8.6
56.2
0
27.3
137
182
0
0
0
43.1
0
28
0
88
14.5
205
HBr
0
5
5
7.1
0
5.5
0
2.6
0
8
4.1
0
0
5.3
6.6
0
1.7
0
15.6
0
5.3
0
0
0
0
0
0
0
0
0
0
0
0
9.6
20.5
0
0
7.1
0
0
5.1
0
NO2
0.26
1.07
0.08
0.43
0.25
0.33
0.37
0.39
0.2
0.63
0.19
0.15
0.09
2
0.15
0.62
0.04
0.04
0.08
0.85
0.29
0.02
0.02
0.03
0.04
0.02
0
0
0.02
0.01
0
0.52
0
1.6
0
0.53
0
1
0.57
0.03
0.39
0.04
SO2
0
0
0
0
0
0
0
0
0
0
0
0.4
1.2
0
0
0
0
0
0
0
0
0
0.7
0
0
0
0
2.2
0
0.9
16.6
2.1
0.3
8.5
1.5
11.2
0
1.8
0
4.8
0.9
4.9
(continued)
7.4
Exercises
303
Table 7.22 (continued)
Material
M4
M5
M5
M5
M5
M6
M6
M6
M6
M6
M6
M7
M7
M7
M7
M7
M7
M7
M7
M7
M7
M8
M8
M8
M8
M8
M8
M8
M8
Time
7.57
6.97
7.47
10.7
13.71
4.94
5.26
5.53
7.46
9.84
10.9
3.7
3.8
3.83
4.04
5.19
6.01
7.56
9.41
9.59
10.79
3.7
3.99
6.56
7.68
9.16
10.33
12.26
14.96
1000/Time
132.1
143.5
133.9
93.5
72.9
202.4
190.1
180.8
134
101.6
91.7
270.3
263.2
261.1
247.5
192.7
166.4
132.3
106.3
104.3
92.7
270.3
250.6
152.4
130.2
109.2
96.8
81.6
66.8
CO
92
114
103
70
56
94
55
46
77
52
41
398
345
406
342
196
148
86
54
55
55
0
90
37
66
45
62
31
9
HCN
0
0
0
0
0
6.7
14.9
13.5
3.1
4.1
2.4
0
0
0
0
0
0
0
2.2
1.7
4.1
15
8.6
3.1
0
0
0
2.7
0
H2S
0.3
0
0
0
0
0
5.3
6.1
0
0.7
0
0
0
0
0
0
0.2
0
0
0
0
0
0
0
0
0
0
0
0
HCl
536
114
221
259
220
0
21.9
24.9
158
19
82
0
0
0
23
0
387
0
197
321
162
0
88
27.7
105
0
61
0
0
HBr
0
0
0
0
0
0
0
0
0
0
0
21
15.5
47
10.3
0
0
47
0
0
0
0
0
0
0
0
0
0
0
NO2
0.01
0
0
0.02
0.01
0.32
0
0
0.04
0.01
0
0
0.01
0
0.04
0
0.01
0
0
0
0.02
0.34
0.59
0.01
0
0.01
0.01
0.22
0.01
SO2
3
0
0
1.4
0.9
0
2.2
2.5
0
1.4
0
0
0
0
0
0
1.9
0
2.6
1.1
2.9
0
0
0
0
0
0
0
0
Exercise 7.4.2 Cockroaches are responsible for 80% of infestations in spaces used
by humans. They associate with humans and have the ability to contaminate food
with their feces and secretions, having both medical and economic implications.
Different insecticides have been formulated, mainly synthetic, and, in some cases,
have led to the development of cockroaches’ resistance. This example deals with the
study of survival in days ( y) of this insect when exposed to two promising fungi in
the biological control of this insect plus an already known control. The data for this
example are shown below (Table 7.23):
304
7
Time of Occurrence of an Event of Interest
Table 7.23 Results of the cockroach biological control experiment
Insect
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
Strain
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Age
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
Time
2
3
3
3
4
7
8
9
10
10
17
19
19
20
20
20
20
20
20
20
4
13
13
13
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
3
3
Insect
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
Strain
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Age
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
Time
2
2
2
3
3
3
3
4
5
8
9
10
11
20
20
20
20
20
20
20
3
3
4
6
8
13
17
17
19
19
20
20
20
20
20
20
20
20
20
20
2
3
Insect
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
Strain
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Age
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
Time
2
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
11
20
(continued)
7.4
Exercises
305
Table 7.23 (continued)
Insect
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Strain
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Bb1
Age
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Time
4
4
6
7
8
8
9
10
13
14
16
17
17
20
20
20
20
20
Insect
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Strain
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Bb2
Age
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Time
4
5
5
10
10
10
11
13
13
13
14
15
15
15
15
19
20
20
Insect
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Strain
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Test
Age
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Time
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
(a)
(b)
(c)
(d)
Write down a statistical model of this experiment.
List all components of the GLMM from (a).
Write down the null and alternative hypotheses associated with this experiment.
Analyze the survival time of the insect when infected with the different types of
fungi.
(e) Comment on the results obtained.
Exercise 7.4.3 Consider a study on the effect of analgesic treatments (Trt) in elderly
patients with neuralgia. Two test treatments (A and B) and a placebo (P) are
compared. The response variable is whether the patient reported pain or not
(yes = 1, n = 0). The investigators recorded the age (E) and sex (S) of 60 patients
and the duration (time = T) in which the pain disappeared after starting the
treatment. The data are presented in the Table 7.24 below.
(a) List all components of the GLMM for this exercise.
(b) Write down the null and alternative hypotheses associated with this experiment.
(c) Construct an ANOVA table indicating the sources of variation and degrees of
freedom.
(d) Analyze the average time during which the patient experiences pain after starting
the treatment. Are there any significant differences?
(e) Comment on the results obtained.
7 Time of Occurrence of an Event of Interest
306
Table 7.24 Results with neuralgia patients (Trt = Treatment, S = Sex, E = Age, T = Time,
D = Pain with yes = 1 and no = 0)
Trt
P
P
A
A
B
A
P
A
B
B
A
B
P
P
A
P
B
P
P
A
S
F
M
F
M
F
F
M
F
F
M
M
M
M
M
M
F
F
M
F
F
E
68
66
71
71
66
64
70
64
78
75
70
70
78
66
78
72
65
67
67
74
T
1
26
12
17
12
17
1
30
1
30
12
1
12
4
15
27
7
17
1
1
D
0
1
0
1
0
0
1
0
0
1
0
0
1
1
1
0
0
1
1
0
Tr
B
B
B
A
A
P
B
A
P
P
A
B
B
P
B
P
P
B
A
B
S
M
F
F
F
M
M
M
M
M
M
F
M
M
F
M
F
F
M
M
M
E
74
67
72
63
62
74
66
70
83
77
69
67
77
65
75
70
68
70
67
80
T
16
28
50
27
42
4
19
28
1
29
12
23
1
29
21
13
27
22
10
21
D
0
0
0
0
0
0
0
0
1
1
0
0
1
0
1
1
1
0
0
1
Tr
P
B
B
A
P
A
B
A
B
P
B
A
B
P
A
A
P
A
P
A
S
F
F
F
F
F
F
M
M
F
F
F
M
F
M
F
M
M
M
F
F
E
67
77
76
69
64
72
59
69
69
79
65
76
69
60
67
75
68
65
72
69
T
30
16
9
18
1
25
29
1
42
20
14
25
24
26
11
6
11
15
11
3
D
0
0
1
1
1
0
0
0
0
1
0
1
0
1
0
1
1
0
1
0
Exercise 7.4.4 Refer to the previous exercise and perform an analysis of
covariance.
(a) List the linear predictor of this experiment.
(b) Analyze the average time during which the patient experiences pain after starting
the treatment using an analysis of covariance. Are there any significant
differences?
(c) Comment on the results obtained. Your results differ from those obtained in the
previous year.
Appendix 1
Data: Onset and duration of estrus in Pelibuey ewes (age in weeks, weight in kilograms,
Inestro = number of days from the onset of estrus, Durestro = number of days in the duration of
estrus)
Animal
1
2
3
4
Birth type
1
1
1
1
Age
18.5096
18.4438
19.3973
19.3973
Weight
52.5
47.4
50.2
53.6
CC
4
4
4
4
Inestro
28
28
16
28
Durestro
4
4
20
16
(continued)
Appendix 1
Animal
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
307
Birth type
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Age
9.4356
18.674
20.0877
19.5616
19.5288
19.7589
29.9507
19.3644
19.0027
18.3452
20.0877
40.7671
51.189
40.0767
54.3123
52.274
53.6219
40.2082
36.4932
50.6301
51.0247
46.389
50.7945
30.411
30.5096
50.6959
36.6247
30.5425
36.6247
29.9507
47.211
40.2082
52.2411
53.4247
55.5616
30.5425
29.0959
40.1425
50.7945
37.6767
36.4274
30.5425
Weight
47.5
41.3
60.5
49.4
53.4
52.6
35.5
50.5
62.2
54.7
48.7
40.5
49.3
38.1
41.9
58
34.8
40.3
34.6
42.1
52.6
32.1
40
37.9
42.2
33.2
34.2
39
33.7
32.9
39.5
57.5
53.3
43.4
46
31.6
47.8
36
42.2
44.3
43.1
38
CC
4
3
5
4
4
5
3
4
5
4
4
3
4
3
3
4
3
2
2
2
4
2
2
2
3
2
3
2
2
2
2
5
4
3
3
2
3
2
3
3
2
2
Inestro
28
28
28
28
28
28
28
28
28
28
28
28
12
20
24
28
28
24
28
28
28
20
16
24
20
24
20
12
24
24
32
12
12
12
24
24
20
20
24
24
20
20
Durestro
4
4
4
4
4
4
4
4
4
4
4
4
8
20
8
4
4
8
4
4
4
12
16
8
20
16
20
32
16
16
12
32
28
32
16
16
20
20
16
16
20
20
(continued)
308
Animal
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
7
Birth type
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Age
68.9753
63.1233
68.7781
64.3068
68.6795
62.6301
69.8959
69.6
63.4849
64.274
63.5178
78.7397
64.537
62.4329
67.6603
63.7151
74.4986
63.6493
72.9205
69.4027
69.9616
69.6
63.4849
63.6164
67.8575
63.6822
65.7534
67.989
61.1836
63.3534
79.8904
63.7151
Time of Occurrence of an Event of Interest
Weight
42.9
44
38.5
48
40.1
46.3
32.5
42.8
51.3
47.7
44.5
38
52.5
41.2
50.8
48.2
33.3
45.1
33
40.4
43.3
43.2
51
57.4
43
49.7
40.1
33.4
51.6
43.3
44.7
37.9
CC
2
4
3
4
2
3
2
3
4
3
3
2
4
2
4
3
2
3
3
3
3
2
4
4
3
4
3
1
4
3
3
3
Inestro
24
24
20
24
20
32
20
20
24
24
12
12
12
12
20
20
32
24
24
24
12
24
24
24
24
24
24
20
20
20
24
20
Durestro
8
8
12
8
12
4
12
12
8
16
28
28
28
28
20
24
8
16
20
16
28
16
16
16
16
16
16
20
20
20
16
20
Data: Beetles
Dose
Low
Low
Low
Low
Medium
Medium
Medium
Medium
High
Insecticide
Insec1
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Insec1
Rep
1
1
1
1
1
1
1
1
1
Frac
0.31
0.82
0.43
0.45
0.36
0.92
0.44
0.56
0.22
Time
3.1
8.2
4.3
4.5
3.6
9.2
4.4
5.6
2.2
(continued)
Appendix 1
Dose
High
High
High
Low
Low
Low
Low
Medium
Medium
Medium
Medium
High
High
High
High
Low
Low
Low
Low
Medium
Medium
Medium
Medium
High
High
High
High
Low
Low
Low
Low
Medium
Medium
Medium
Medium
High
High
High
High
309
Insecticide
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Insec1
Insec2
Insec3
Insec4
Rep
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
Frac
0.3
0.23
0.3
0.45
1.1
0.45
0.71
0.29
0.61
0.35
1.02
0.21
0.37
0.25
0.36
0.46
0.88
0.63
0.66
0.4
0.49
0.31
0.71
0.18
0.38
0.24
0.31
0.43
0.72
0.76
0.62
0.23
1.24
0.4
0.38
0.23
0.29
0.22
0.33
Time
3
2.3
3
4.5
11
4.5
7.1
2.9
6.1
3.5
10.2
2.1
3.7
2.5
3.6
4.6
8.8
6.3
6.6
4
4.9
3.1
7.1
1.8
3.8
2.4
3.1
4.3
7.2
7.6
6.2
2.3
12.4
4
3.8
2.3
2.9
2.2
3.3
310
7
Time of Occurrence of an Event of Interest
Data: Rubber break time
Block
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
Demineralization
2
2
2
1
1
1
1
1
1
2
2
2
2
2
2
1
1
1
1
1
1
2
2
2
Pasteurization
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
pH
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
y
198.5
299
223.1
166.6
196.5
178.9
160.7
151.1
146.5
146.3
169.3
198.1
236.3
330.7
281.2
151.8
156
159.7
171.8
159.3
155.9
175.2
182.2
136.2
Data: Aedes aegypti (Trt = treatment, Rep = repetition, Y = survival time)
Trt
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Rep
1
1
1
1
1
1
1
1
1
1
2
2
2
2
Y
8
11
11
11
11
11
13
13
14
20
8
11
11
11
Trt
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Rep
1
1
1
1
1
1
1
1
1
1
2
2
2
2
Y
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Trt
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
Rep
1
1
1
1
1
1
1
1
1
1
2
2
2
2
Y
3
3
3
3
4
5
6
6
9
12
3
3
3
3
Trt
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
Rep
1
1
1
1
1
1
1
1
1
1
2
2
2
2
Y
2
2
2
2
3
3
3
3
3
4
2
2
2
2
Trt
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Rep
1
1
1
1
1
1
1
1
1
1
2
2
2
2
Y
2
2
2
2
3
3
3
3
6
12
2
2
2
3
(continued)
Appendix 1
Trt
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Control
Rep
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
311
Y
11
11
15
15
15
16
11
11
11
11
23
25
26
27
30
30
8
8
11
13
14
19
20
20
20
22
Trt
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Rep
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
Y
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Trt
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
Rep
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
Y
3
3
3
4
5
6
3
3
3
2
2
5
5
6
10
12
3
3
3
4
4
5
5
6
9
12
Trt
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
Rep
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
Y
2
2
3
3
3
4
2
2
2
2
3
3
3
3
4
4
2
2
2
2
2
2
3
3
3
3
Trt
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Rep
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
Y
3
3
3
4
4
4
2
2
2
2
3
3
3
3
4
4
2
2
2
3
3
3
4
5
6
11
Data: Aedes aegypti (Bio = bioassay, Trt = treatment, Rep = repetition, Y = survival time)
Bio
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
Trt
C
C
C
C
C
C
C
C
C
C
C
C
C
C
Rep
1
1
1
1
1
1
1
1
1
1
2
2
2
2
Y
8
11
11
11
11
11
13
13
14
20
8
11
11
11
Bio
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
Trt
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
Rep
3
3
3
3
3
3
3
3
3
3
4
4
4
4
Y
3
3
3
2
2
5
5
6
10
12
3
3
3
4
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
C
C
C
C
C
C
C
C
C
C
C
C
C
C
Rep
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Y
5
7
8
8
10
13
14
16
20
22
22
23
23
23
(continued)
312
Bio
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
7
Trt
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Rep
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
Y
11
11
15
15
15
16
11
11
11
11
23
25
26
27
30
30
8
8
11
13
14
19
20
20
20
22
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Bio
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
Trt
MaS
MaS
MaS
MaS
MaS
MaS
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
Time of Occurrence of an Event of Interest
Rep
4
4
4
4
4
4
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
Y
4
5
5
6
9
12
2
2
2
2
3
3
3
3
3
4
2
2
2
2
2
2
3
3
3
4
2
2
2
2
3
3
3
3
4
4
2
2
2
2
2
2
3
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
Rep
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Y
24
24
24
24
28
28
10
11
11
12
12
15
15
16
16
16
16
18
19
27
27
27
27
27
28
28
16
19
19
19
19
19
25
25
26
28
28
28
28
28
28
28
28
(continued)
Appendix 1
Bio
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
Trt
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
313
Rep
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
Y
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
4
5
6
6
9
12
3
3
3
3
3
3
3
4
5
6
Bio
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
B1
Trt
MaC
MaC
MaC
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Rep
4
4
4
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
Y
3
3
3
2
2
2
2
3
3
3
3
6
12
2
2
2
3
3
3
3
4
4
4
2
2
2
2
3
3
3
3
4
4
2
2
2
3
3
3
4
5
6
11
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Rep
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Y
28
28
28
16
17
17
17
17
17
17
19
28
28
28
28
28
28
28
28
28
28
28
28
2
3
3
4
4
4
5
5
5
6
6
6
7
15
17
21
25
28
28
28
(continued)
314
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
7
Trt
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Rep
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
Y
2
2
2
3
3
3
4
7
11
11
11
13
13
14
14
14
15
16
16
23
3
3
5
6
8
8
10
10
11
11
11
12
17
17
17
17
17
17
18
25
4
4
5
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaC
MaC
Time of Occurrence of an Event of Interest
Rep
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
1
1
Y
18
5
5
5
9
10
10
10
12
12
12
12
12
14
15
18
18
23
25
25
25
5
5
6
6
6
6
7
8
8
9
10
10
10
11
11
11
11
11
11
24
2
2
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Rep
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
Y
13
23
2
2
3
6
6
6
8
8
8
8
9
9
9
9
10
10
12
19
20
24
2
2
3
3
3
4
4
5
5
5
6
6
6
7
8
8
8
10
17
21
2
(continued)
Appendix 1
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
Mam
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
315
Rep
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
Y
7
9
10
12
12
12
12
12
12
13
13
13
18
27
27
27
27
2
2
2
2
3
3
3
3
4
5
5
5
6
6
8
8
9
11
13
21
3
3
4
6
6
8
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
Rep
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
Y
2
2
2
2
3
3
3
3
3
4
4
5
5
7
7
9
9
10
2
2
2
3
3
3
3
3
5
6
6
6
7
7
7
9
10
10
18
19
2
3
6
6
8
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Rep
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
Y
2
2
3
3
3
4
5
6
7
7
7
8
9
9
10
10
10
10
10
2
3
3
3
5
7
7
7
8
8
8
9
10
10
10
10
10
10
11
17
4
5
8
8
(continued)
316
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
7
Trt
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
MaS
Rep
2
2
2
2
2
2
2
2
2
2
2
2
2
Y
8
9
9
10
10
11
11
11
11
11
12
12
12
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
MaC
Time of Occurrence of an Event of Interest
Rep
3
3
3
3
3
3
3
3
3
3
3
3
3
Y
8
9
9
9
9
9
10
10
10
10
10
11
13
Bio
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
B2
Trt
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Ma1
Rep
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
Y
8
9
9
11
11
11
11
11
12
12
13
13
13
13
14
18
Data: Pelibuey Sheep
Animal
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
Birthtype
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
Age
18.509589
18.4438356
19.3972603
19.3972603
9.43561644
18.6739726
20.0876712
19.5616438
19.5287671
19.7589041
29.9506849
19.3643836
19.0027397
18.3452055
20.0876712
40.7671233
51.1890411
40.0767123
54.3123288
52.2739726
53.6219178
40.2082192
36.4931507
50.630137
Weight
52.5
47.4
50.2
53.6
47.5
41.3
60.5
49.4
53.4
52.6
35.5
50.5
62.2
54.7
48.7
40.5
49.3
38.1
41.9
58
34.8
40.3
34.6
42.1
Inestro
28
28
16
28
28
28
28
28
28
28
28
28
28
28
28
28
12
20
24
28
28
24
28
28
Durestro
4
4
20
16
4
4
4
4
4
4
4
4
4
4
4
4
8
20
8
4
4
8
4
4
(continued)
Appendix 1
Animal
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
317
Birthtype
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Age
51.0246575
46.3890411
50.7945206
30.4109589
30.509589
50.6958904
36.6246575
30.5424658
36.6246575
29.9506849
47.2109589
40.2082192
52.2410959
53.4246575
55.5616438
30.5424658
29.0958904
40.1424658
50.7945206
37.6767123
36.4273973
30.5424658
68.9753425
63.1232877
68.7780822
64.3068493
68.6794521
62.630137
69.8958904
69.6
63.4849315
64.2739726
63.5178082
78.739726
64.5369863
62.4328767
67.660274
63.7150685
74.4986301
63.6493151
72.920548
69.4027397
69.9616438
Weight
52.6
32.1
40
37.9
42.2
33.2
34.2
39
33.7
32.9
39.5
57.5
53.3
43.4
46
31.6
47.8
36
42.2
44.3
43.1
38
42.9
44
38.5
48
40.1
46.3
32.5
42.8
51.3
47.7
44.5
38
52.5
41.2
50.8
48.2
33.3
45.1
33
40.4
43.3
Inestro
28
20
16
24
20
24
20
12
24
24
32
12
12
12
24
24
20
20
24
24
20
20
24
24
20
24
20
32
20
20
24
24
12
12
12
12
20
20
32
24
24
24
12
Durestro
4
12
16
8
20
16
20
32
16
16
12
32
28
32
16
16
20
20
16
16
20
20
8
8
12
8
12
4
12
12
8
16
28
28
28
28
20
24
8
16
20
16
28
(continued)
318
Animal
22
23
24
25
26
27
28
29
30
31
32
7
Birthtype
3
3
3
3
3
3
3
3
3
3
3
Age
69.6
63.4849315
63.6164384
67.8575343
63.6821918
65.7534247
67.9890411
61.1835616
63.3534247
79.890411
63.7150685
Time of Occurrence of an Event of Interest
Weight
43.2
51
57.4
43
49.7
40.1
33.4
51.6
43.3
44.7
37.9
Inestro
24
24
24
24
24
24
20
20
20
24
20
Durestro
16
16
16
16
16
16
20
20
20
16
20
Data: Gum Breakdown Times
Batch
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
Demineralization
2
2
2
1
1
1
1
1
1
2
2
2
2
2
2
1
1
1
1
1
1
2
2
2
Pasteurization
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
pH
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
Time
198.5
299
223.1
166.6
196.5
178.9
160.7
151.1
146.5
146.3
169.3
198.1
236.3
330.7
281.2
151.8
156
159.7
171.8
159.3
155.9
175.2
182.2
136.2
Appendix 1
319
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Chapter 8
Generalized Linear Mixed Models
for Categorical and Ordinal Responses
8.1
Introduction
According to Agresti (2013), a multinomial distribution is a generalization of a
binomial distribution in cases with more than two possible ordered (ordinal) or
unordered (nominal) outcomes. Given a response with more than two possible
outcomes and independent trials with probabilities of similar category for each
trial, the distribution of counts across categories follows a multinomial distribution.
Quinn and Keough (2002) believe that several methods exist for multinomial data
analysis. The most common form of categorical data analysis in biological sciences,
which results in frequency counts, is creating cross-tabulations or contingency tables
and chi-squared tests to examine associations between two or more categorical
variables. However, such an approach is ill suited for a study aimed at estimating
the response when there is a change in the explanatory variable(s), as contingency
tables are used to analyze the association between variables without considering a
predictor or response variable. In this analysis, the results are valid as long as less
than 20% of the cells have an expected count less than five and none are less than one
(Logan 2010). Fisher’s exact test extends the chi-squared test in studies involving
small sample sizes.
There are several methods for modeling multinomial data; traditional methods of
multinomial data analysis include frequency analysis (counts), which uses the
chi-squared test and the log-linear model for contingency tables. This chapter
focuses on describing multinomial logit and probit models in detail.
© The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8_8
321
322
8.2
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Concepts and Definitions
For the multinomial distribution each observation drawn from a total of
N observations belongs to exactly one of the mutually and exclusive c = 1, ⋯,
C categories and each category has a probability π c (c = 1, ⋯, C) of belonging to the
category c. A multinomial distribution refers to the probability that exactly one
randomly sampled observation from the population belongs to category y1, that is, it
belongs to category 1, y2 observations belong to category 2, and so forth up to
C
category C,where
c=1
yc = N and
C
c=1
π c = 1. The density function of this distribution
is equal to
f ð y1 , y 2 , . . . , y C Þ =
N!
y
y y
π 1 π 2 . . . π Cc
y1 !y2 ! . . . yC ! 1 2
Multinomial models are applied in data analysis where the categorical response
variable has more than two possible outcomes while the independent variables can
be continuous, categorical, or both (Hosmer and Lemeshow 2000). The categorical
response variable can be either ordinal (ordered) or nominal (unordered). Ordinal
response variables are single values that represent a rank order on some dimension,
but there are not enough values to be treated as a continuous variable. Nominal
(unordered) response variables are those whose values provide a rank but do not
provide an indication of order. Models for multinomial data are constructed in a
similar way as for binomial data. The link functions used in these types of models are
similar to the logit and probit functions used for binomial data. Cumulative logit and
cumulative probit models define the link function such that when properly fitted to
the data, they allow for parsimonious modeling of ordinal or multinomial data.
Generalized logit and probit models do not require ordered categories and are
therefore suitable for multinomial nominal data.
In terms of generalized linear models (GLMs) and generalized linear mixed
models (GLMMs), a multinomial distribution with C categories requires C - 1
link functions to fully specify a model that relates the response probabilities
(π 1, π 2, . . ., π C) to the linear predictor. The commonly used models are the cumulative logit model, also known as the proportional odds model proposed by McCullagh
(1980), and the cumulative probit model, also known as the threshold model.
Throughout this chapter, we will use either of these two link functions
interchangeably.
The link functions for a cumulative logit model with C categories are
8.3
Cumulative Logit Models (Proportional Odds Models)
η1 = log
323
π1
= η1 þ Xβ þ Zb
1 - π1
π1 þ π2
= η2 þ Xβ þ Zb
1 - ðπ 1 þ π 2 Þ
⋮
π1 þ π2 þ ⋯ þ πC - 1
ηC - 1 = log
= ηC - 1 þ Xβ þ Zb
1 - ðπ 1 þ π 2 þ ⋯ þ π C - 1 Þ
η2 = log
where X and Z are the design matrices, whereas β and b are the vectors of fixed and
random effects parameters, respectively. The inverse links of each of the functions
are as follows:
1
= hðη1 Þ
1 þ e - η1
1
= hðη2 Þ
π1 þ π2 =
1 þ e - η2
⋮
1
π1 þ π2 þ ⋯ þ πC - 1 =
= hðηC - 1 Þ:
1 þ e - ηc - 1
π1 =
Once h(η1), h(η2), ... h(ηc
probabilities π^1 , π^2 , ..., π^c .
8.3
- 1)
have been estimated, we can then estimate the
Cumulative Logit Models (Proportional Odds Models)
Multinomial logit models are used to model the relationships between a polytomous
response variable and a set of predictor variables. These polytomous response
models can be classified – as mentioned above – into two different types, depending
on whether the response variable has an ordered or an unordered structure.
In a proportional odds model, the covariates (linear predictor η) have the same
effect on the probabilities that the response variable has in any category when
considering different values of the covariates, thus shifting the response distribution
to the right (or left) without changing the shape of the distribution. In a proportional
odds model, the cumulative logits model the effect of the covariates on the response
probabilities below or equal to the category cutoff.
A multinomial logit model assumes independence of categories, which implies
that the probabilities of choosing a category c relative to a category c′ are independent of the category characteristics of c and c′ for c ≠ c′. The assumption requires that
if a new category is available, then the prior probabilities are precisely adjusted to
preserve the original probabilities between all pairs of outcomes. The proportional
odds model employs a strict assumption that the odds ratio does not depend on the
category, and, therefore, we need to test the proportional odds assumption, which is
also called the “parallel regression assumption.”
324
8.3.1
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Complete Randomize Design (CRD) with a Multinomial
Response: Ordinal
Data are obtained from an experiment related to red core disease in strawberries,
which is caused by the fungus Phytophthora fragariae. In this example, 12 strawberry populations were evaluated in a completely randomized experiment with
4 replications (Table 8.1). Plots generally consisted of 10 plants; in some cases,
only 9 plants were observed. At the end of the experiment, each plant was assigned
to one of three ordered categories representing fungal damage (1 = no damage,
2 = moderate damage, and 3 = severe damage).
A total of 12 populations were obtained by crossing 3 genotypes of male parents
with 4 genotypes of female parents. The variation between and within plots is
considered minimal, whereas the genetic and nongenetic effects are more significant,
as plants from the same cross are not genetically identical.
The model that fits these data for the cumulative probabilities is a GLMM, which
exhibit a classification effect on the treatment variable (population resulting from
crossing genotypes). Thus, the GLMM for multinomial ordered outcomes with
C categories requires C - 1 link function equations to fully specify the model that
relates the response probabilities (π 1, π 2, . . ., π C) to the linear predictor ηij (Stroup
2013). The C - 1 multinomial logit equations are tested against each of the
remaining categories 1, 2, . . , C - 1.
Table 8.1 Evaluation of red core disease in strawberry plants
Repetition
Parent plant male/female
1
1
1
2
1
3
1
4
2
1
2
2
2
3
2
4
3
1
3
2
3
3
3
4
1
2
Disease category
1
2
3
1
0
3
6
2
2
3
5
0
3
4
3
7
0
5
5
5
1
4
4
2
1
4
5
3
4
3
3
5
1
4
5
1
0
0
9
3
5
3
2
3
0
3
6
2
3
0
7
5
3
2
2
3
2
4
2
4
1
2
5
2
5
2
3
6
7
1
1
6
2
4
6
2
5
3
3
1
2
4
1
2
1
1
3
8
2
3
1
7
4
2
3
6
1
8
2
6
3
5
5
6
3
3
3
5
0
7
0
7
3
4
0
3
1
6
0
1
2
2
2
1
1
4
4
2
0
2
0
3
2
5
3
3
4
5
2
2
5
0
1
3
4
3
3
5
5
5
4
4
4
3
10
7
7
3
8.3
Cumulative Logit Models (Proportional Odds Models)
325
The components of the GLMM with an ordinal multinomial response are as
follows:
Distributions: y1ij, y2ij, y3ij|rj~Multinomial(Nij, π 1ij, π 2ij, π 3ij), where y1ij, y2ij, and y3ij
are the observed frequencies of responses (damage level) in each category
C (1 = no damage, 2 = moderate damage, and 3 = severe damage) and rj is
the random effect due to repetition, assuming r j N 0, σ 2r .
Linear predictor: ηcij = ηc + τi + rj, where ηcij is the cth link (c = 1, 2, 3) that relates
the mean and the linear predictor for the treatment i (i = 1, 2, . . ., 12) and the jth
block ( j = 1, 2, 3, 4); ηc is the intercept for the cth link; τi is the fixed effect due to
the ith treatment (cross); and rj is the random effect due to the jth repetition
r j N 0, σ 2r . The link functions for each category are as follows:
log
log
π 1ij
= η1ij
1 - π 1ij
π 1ij þ π 2ij
1 - π 1ij þ π 2ij
= η2ij
The following GLIMMIX program fits a cumulative logit model with an ordinal
multinomial response in a CRD.
proc glimmix data=FRESA;
class rep trt cat;
model cat(order=data)= trt/dist=Multinomial link=clogit solution
oddsratio;
random intercept/subject=rep solution ;
estimate 'c=1, t=1' intercept 1 0 trt 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,
'c=2, t=1' intercept 0 1 trt 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,
'c=1, t=2' intercept 1 0 trt 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,
'c=2, t=2' intercept 0 1 trt 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,
'c=1, t=3' intercept 1 0 trt 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0,
'c=2, t=3' intercept 0 1 trt 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0,
'c=1, t=4' intercept 1 0 trt 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0,
'c=2, t=4' intercept 0 1 trt 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0,
'c=1, t=5' intercept 1 0 trt 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0,
'c=2, t=5' intercept 0 1 trt 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0,
'c=1, t=6' intercept 1 0 trt 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0,
'c=2, t=6' intercept 0 1 trt 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0,
'c=1, t=7' intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0,
'c=2, t=7' intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0,
'c=1, t=8' intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0,
'c=2, t=8' intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0,
'c=1, t=9' intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0,
'c=2, t=9' intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0,
'c=1, t=10' intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0,
'c=2, t=10' intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0,
326
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.2 Data ordered
Rep
rep1
rep1
rep1
rep1
rep1
rep1
rep1
rep1
rep1
rep1
rep1
Cross
M1H1
M1H2
M1H3
M1H4
M1H1
M1H2
M1H3
M1H4
M1H1
M1H2
M1H3
Cat
Without
Without
Without
Without
Moderate
Moderate
Moderate
Moderate
Severe
Severe
Severe
Freq
0
2
3
0
3
3
4
5
6
5
3
'c=1, t=11' intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0,
'c=2, t=11' intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0,
'c=1, t=12' intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1,
'c=2, t=12' intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/ilink;
freq freq;
run;
Although most of the GLIMMIX commands have already been described in
previous examples, it is important to emphasize that the data should be structured
in a logical way as follows: one line for repetition, treatment, lesion category, and the
frequency or number of observations (Y ), which, in this case, is referenced by the
variables rep, trt (trt = cross), cat (category), and freq, respectively. Part of the data
arrangement can be seen in Table 8.2, whereas the rest of the dataset can be found in
the Appendix (Data: CRD with multinomial response: ordinal).
In the program commands of this example, “order = data” indicates that the order
in which the categories are arranged in the dataset is under an order (ordinal)
category. Consider that the observations in each line always have order categories
such as no injury (Without), moderate injury (Moderate), and severe injury (Severe).
If there is no congruent order in the arrangement of the dataset to be analyzed, then
GLIMMIX will reorder the categories in an alphabetical or numerical order
depending on the initial coding of the data. The “estimate” command specifies the
estimable functions that form the boundaries between the categories for each of the
populations (trt). Finally, the “freq command” instructs GLIMMIX to use “freq” as
the number of observations (frequency) under the corresponding categorization. In
this way, the first estimate “c = 1, t = 1” defines the predictor η1 + τ1, that is, the
boundary between the “Without” and “Moderate” categories for treatment 1 with its
corresponding logit log 1 -π 11π11 , whereas the second estimate “c = 2, t = 1” defines
the boundary between the categories of “Moderate” and “Severe” damage with the
21
, which estimates the probability of observing a plant from
logit log 1 -πð11πþπ
11 þπ 21 Þ
8.3
Cumulative Logit Models (Proportional Odds Models)
327
Table 8.3 Results of the multinomial analysis of variance for injury level in strawberry plants
(a) Covariance parameter estimates
Cov Parm
Subject
Intercept
Rep
(b) Type III tests of fixed effects
Effect
Num degree of freedom (DF)
Trt
11
Estimate
0.1453
Den DF
457
Standard error
0.1437
F-value
2.60
Pr > F
0.0032
Table 8.4 Fixed effects solution for injury categories
Solutions for fixed effects
Effect
Cat
Intercept Without ðη1 Þ
Intercept Moderate ðη2 Þ
ðτ 1 Þ
Trt
ðτ 2 Þ
Trt
ðτ 3 Þ
Trt
Trt
ðτ 4 Þ
ðτ 5 Þ
Trt
ðτ 6 Þ
Trt
ðτ 7 Þ
Trt
Trt
ðτ 8 Þ
ðτ 9 Þ
Trt
ðτ10 Þ
Trt
ðτ11 Þ
Trt
Trt
ðτ12 Þ
Trt
M1H1
M1H2
M1H3
M1H4
M2H1
M2H2
M2H3
M2H4
M3H1
M3H2
M3H3
M3H4
Estimate
-0.4571
1.0631
-1.1456
-0.8355
-0.4621
-0.4716
-1.2644
-0.6060
-0.2332
-0.3912
-1.5563
-0.4508
-1.4426
0
Standard error
0.3526
0.3558
0.4264
0.4179
0.4171
0.4145
0.4295
0.4181
0.4140
0.4168
0.4393
0.4144
0.4350
.
DF
3
3
457
457
457
457
457
457
457
457
457
457
457
.
t-value
-1.30
2.99
-2.69
-2.00
-1.11
-1.14
-2.94
-1.45
-0.56
-0.94
-3.54
-1.09
-3.32
.
Pr > |t|
0.2855
0.0582
0.0075
0.0462
0.2685
0.2558
0.0034
0.1479
0.5735
0.3484
0.0004
0.2772
0.0010
.
population1 (M1H1 = trt) “Without” damage and “Moderate” damage when
exposed to the fungus (Phytophthora fragariae). Part of the output is presented in
Table 8.3.
The estimated variance component (part (a)) due to plants is σ 2r = 0:1453,
whereas the hypothesis tests for type III effects (part (b)) (“Type III tests of fixed
effects”) indicate that the crosses have different significant tolerance levels to fungal
attacks (Pr > F = P = 0.0032). The results of the fixed effects solution, obtained by
specifying the “solution” option in the model, are shown in Table 8.4.
From the fixed effects solution, we can estimate the linear predictors for the two
categories of each treatment, which are in terms of the model scale. For example,
for treatment 1, the first category of injury η11 = η1 þ τ1 = - 0:4571 þ
ð- 1:1456Þ = - 1:6027, where η1 defines the boundary between the categories
“Without” damage and “Moderate” damage and η2 defines the boundary between
the categories “Moderate” damage and “Severe” damage, and the linear predictor is
η11 = η2 þ τ1 = 1:0631 þ ð- 1:1456Þ = - 0:0825. Note that for the proportional
odds, the τi values are not category-specific; treatment effects move the boundaries
as a group.
328
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.5 Estimated odds
ratio
Odds ratio estimates
Trt
_Trt
Estimate
M1H1 M3H4 0.318
M1H2 M3H4 0.434
M1H3 M3H4 0.630
M1H4 M3H4 0.624
M2H1 M3H4 0.282
M2H2 M3H4 0.546
M2H3 M3H4 0.792
M2H4 M3H4 0.676
M3H1 M3H4 0.211
M3H2 M3H4 0.637
M3H3 M3H4 0.236
DF
457
457
457
457
457
457
457
457
457
457
457
95% Confidence limits
0.138
0.735
0.191
0.986
0.278
1.430
0.276
1.409
0.121
0.657
0.240
1.241
0.351
1.787
0.298
1.534
0.089
0.500
0.282
1.438
0.101
0.556
The odds ratio (Table 8.5) is the result of taking eτi for crosses 1–12. Since odds
ratios are not specific to a particular category, this value is the same for all three
categories and hence the name odds ratio.
In Table 8.6, we show the maximum likelihood estimates of the linear predictors
^ηci = ^ηC þ ^τi in the “Estimate” column, in terms of the model scale, as well as the
means on the data scale for each of the categories of the treatments tested (“Mean”).
Thus, for c = 1, t = 1 (response category “Without” damage and treatment 1), the
estimator is η11 = - 1:6027 and for c = 2, t = 1 (“Moderate” damage and treatment
1), the linear predictor is η21 = - 0:0825. Taking the inverse of the link function
yields the probability of π 11 = 1=1þe1:6027 = 0:1676. This is the estimated probability
for which the cross (treatment) M1H1 has a response score of “Without damage.”
This inverse value is presented under the “Mean” column (Table 8.6).
Now, for c = 2, t = 1, the inverse of the link yields the following probability:
π 11 þ π 21 = 1=1þe0:0825 = 0:4794 (cumulative probability). From this value, we deduce
the probability of observing a “Moderate” damage and a “Severe” damage in
the
plant
of
the
cross
M1H1.
For
“Moderate”
damage,
and,
for
the probability is π 21 = 0:4794 - π 11 = 0:4794 - 0:1676 = 0:3118,
“Severe” damage, it is π 31 = 1 - π 11 þ π 21 = 1 - 0:4794 = 0:5206. Similarly, the
rest of the probabilities in the different crosses are estimated.
8.3.2
Randomized Complete Block Design (RCBD)
with a Multinomial Response: Ordinal
In recent years, poultry production has become conscious of animal welfare, which
is associated with bird mortality, behavior, and health, among others (Stanley 1981;
Martrenchar et al. 2002). One of the diseases related to animal welfare is footpad
dermatitis, and, among many repercussions, it affects a bird’s ability to walk (Bilgili
et al. 2009). Pododermatitis is known as contact dermatitis or footpad dermatitis and
8.3
Cumulative Logit Models (Proportional Odds Models)
329
Table 8.6 Estimates on the model scale (Estimate) and on the data scale (Mean) for the damage
categories in strawberry plants
Estimates
Label
c = 1, t = 1
c = 2, t = 1
c = 1, t = 2
c = 2, t = 2
c = 1, t = 3
c = 2, t = 3
c = 1, t = 4
c = 2, t = 4
c = 1, t = 5
c = 2, t = 5
c = 1, t = 6
c = 2, t = 6
c = 1, t = 7
c = 2, t = 7
c = 1, t = 8
c = 2, t = 8
c = 1, t = 9
c = 2, t = 9
c = 1,
t = 10
c = 2,
t = 10
c = 1,
t = 11
c = 2,
t = 11
c = 1,
t = 12
c = 2,
t = 12
Standard
error
0.3706
0.3625
0.3597
0.3542
0.3572
0.3555
0.3542
0.3524
0.3744
0.3656
0.3590
0.3557
0.3526
0.3533
0.3566
0.3556
0.3864
0.3759
0.3540
DF
457
457
457
457
457
457
457
457
457
457
457
457
457
457
457
457
457
457
457
t-value
-4.33
-0.23
-3.59
0.64
-2.57
1.69
-2.62
1.68
-4.60
-0.55
-2.96
1.28
-1.96
2.35
-2.38
1.89
-5.21
-1.31
-2.56
Pr > |t|
<0.0001
0.8200
0.0004
0.5208
0.0104
0.0916
0.0090
0.0939
<0.0001
0.5822
0.0032
0.1995
0.0509
0.0193
0.0178
0.0595
<0.0001
0.1902
0.0106
Mean
0.1676
0.4794
0.2154
0.5567
0.2851
0.6459
0.2832
0.6437
0.1517
0.4499
0.2567
0.6123
0.3340
0.6963
0.2998
0.6619
0.1178
0.3791
0.2874
Standard error
mean
0.05170
0.09047
0.06080
0.08741
0.07281
0.08131
0.07190
0.08081
0.04818
0.09047
0.06850
0.08444
0.07842
0.07471
0.07485
0.07958
0.04016
0.08849
0.07250
0.6123
0.3524
457
1.74
0.0830
0.6485
0.08033
-1.8997
0.3813
457
-4.98
<0.0001
0.1301
0.04317
-0.3795
0.3714
457
-1.02
0.3074
0.4062
0.08958
-0.4571
0.3526
457
-1.30
0.1955
0.3877
0.08369
1.0631
0.3558
457
2.99
0.0030
0.7433
0.06789
Estimate
-1.6027
-0.08254
-1.2926
0.2276
-0.9191
0.6010
-0.9286
0.5915
-1.7214
-0.2013
-1.0631
0.4571
-0.6903
0.8299
-0.8483
0.6719
-2.0133
-0.4932
-0.9079
is characterized by inflammation and necrotic lesions from the plantar surface to
deep within the footpads of chicken. Deep ulcers may result in abscesses and in the
thickening of the underlying tissues and structures (Greene et al. 1985).
Chicken feet have great economic importance because they are in high demand in
the foreign market, mainly in Southeast Asia and China; however, due to diseases or
alterations such as pododermatitis, there are significant economic losses since
diseased feet are not suitable for human consumption and this, subsequently, reflects
in market prices (Taira et al. 2014). Due to the economic importance of this product,
Garcia et al. (2010) have focused on studying the factors that cause this disease and
330
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.7 Treatment design
Treatment
Trt1
Trt2
Trt3
Trt4
Features
Traditional program +1 kg m-2 of rice husks
Traditional program +2 kg m-2 of rice husks
Traditional program + podal health program +1 kg m-2 of rice husks
Traditional program + podal health program +2 kg m-2 of rice husks
on finding strategies to reduce leg and carcass lesions in poultry. Important factors in
broiler fattening are the type of litter, litter height, nutrition and feeding programs,
and bird health, among others.
The objective of this study was to evaluate the effect of litter density and organic
minerals (Availa Zn and Availa Mn), with an extract of Yucca schidigera (MicroAid) as a supplement to a traditional fattening program, on the development of
footpad dermatitis in broilers. The genetic material used in this experiment was
mainly male Ross line chickens. The traditional broiler fattening program by the
poultry farm consists of three phases: a starter diet (1–18 days), a grower diet
(19–35 days), and a finisher diet (36–50 days), applied for a period of 50 days,
where rice husk is used as bedding material at a density of 1 kg m-2. In this research,
a foot health program was implemented in addition to the traditional fattening
program, which included the addition of 125 ppm of Micro-Aid (Yucca schidigera
extract), 40 ppm of Availa Zn, and 40 ppm of Availa Mn to the fattening diet.
Based on the above information, four treatments were evaluated at two poultry
farms, as described below:
• Treatment 1 involved the application of the company’s traditional fattening
program (Trt1).
• Treatment 2 was the company’s traditional fattening program plus an increase in
litter density from 1 to 2 kg m-2 (Trt2).
• Treatment 3 was the traditional fattening program plus the implementation of the
foot health program during the fattening period until completion (Trt3).
• Treatment 4 consisted of the traditional fattening program plus the implementation of the foot health program and an increase in litter density from 1 to 2 kg m-2
(Trt4). The following table lists the treatments studied (Table 8.7):
The response variable evaluated was the degree of foot lesion (pododermatitis) at
the end of the fattening period (50 days). The response variable was evaluated on
1250 chickens per treatment. The degree of a footpad lesion was determined
according to a visual guide for lesions in chickens based on the method of De
Jong and Guémené (2012). This method entails defining three grades: grade 0 is
attributed to legs with no lesions, grade one is if lesions exist in some areas of the
footpad (<50%), and grade two is if the leg has extensive lesions in areas of the
footpad (50–100%). Table 8.8 shows the dataset indicating the block, treatment,
level of lesion, and the number of birds observed with a given lesion (frequency).
8.3
Cumulative Logit Models (Proportional Odds Models)
331
Table 8.8 Pododermatitis in broilers
Block
1
1
1
2
2
2
1
1
1
2
2
2
Trt
1
1
1
1
1
1
2
2
2
2
2
2
Category
Without
Slight
Severe
Without
Slight
Severe
Without
Slight
Severe
Without
Slight
Severe
Frequency
26
58
17
37
56
6
40
57
3
77
23
0
Block
1
1
1
2
2
2
1
1
1
2
2
2
Trt
3
3
3
3
3
3
4
4
4
4
4
4
Category
Without
Slight
Severe
Without
Slight
Severe
Without
Slight
Severe
Without
Slight
Severe
Frequency
54
43
3
25
69
7
65
34
1
63
36
0
Note: Without stands for no lesion, slight stands for moderate lesion, and severe stands for severe
lesion
The GLMM for multinomial ordered results with C categories requires C - 1 link
function equations instead of one to fully specify a model that relates the response
probabilities (π 1, π 2, . . ., π C) to the linear predictor ηij (Stroup 2013). The C - 1
multinomial logit equations are tested against each of the categories 1, 2, . . ., C - 1.
The link functions for the cumulative logit model to describe the response
variable with C categories are as follows:
ηð1Þij = log
ηð2Þij = log
π 1ij
= η1 þ τi þ bj
1 - π 1ij
π 1ij þ π 2ij
1 - π 1ij þ π 2ij
= η 2 þ τ i þ bj
⋮
π 1ij þ π 2ij þ ⋯ þ π ðC - 1Þij
ηðC - 1Þij = log
1 - π 1ij þ π 2ij þ ⋯ þ π ðC - 1Þij
= η C - 1 þ τ i þ bj
The components of the GLMM with an ordinal multinomial response variable are
as follows:
Distributions: yoij, y1ij, y2ij|bj ~ Multinomial(Nij, π 0ij, π 1ij, π 2ij), where yoij, y1ij, and y2ij
are the observed frequencies of the responses (paw injury) in each category (none,
mild, and severe) and bj is the random effect due to block assuming
bj N 0, σ 2b .
Linear predictor: η(c)ij = ηc + τi + bj, where η(c)ij is cth link (c = 0, 1) for processing
i and block j, ηc is the intercept for the cth link, τi is the fixed effect due to the ith
treatment, and bj is the random effect due to the jth block bj N 0, σ 2b . The
link functions for each category are as follows:
332
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
log
log
π 0ij
= ηð0Þij
1 - π 0ij
π 0ij þ π 1ij
1 - π 0ij þ π 1ij
= ηð1Þij
The following GLIMMIX commands fit a cumulative logit model with an ordinal
multinomial response.
proc glimmix data=multinomial_ord;
class block trt;
model categoria (order=data)= trt/dist=Multinomial link=clogit
solution oddsratio(DIFF=LAST LABEL);
random intercept/subject=block;
estimate 'c=0, t=1' intercept 1 0 trt 1 0,
'c=1, t=1' intercept 0 1 trt 1 0,
'c=0, t=2' intercept 1 0 trt 0 1 0,
'c=1, t=2' intercept 0 1 trt 0 1 0,
'c=0, t=3' intercept 1 0 trt 0 0 0 1 0,
'c=1, t=3' intercept 0 1 trt 0 0 0 1 0,
'c=0, t=4' intercept 0 1 trt 0 0 0 0 1,
'c=1, t=4' intercept 1 0 trt 0 0 0 0 1/ilink;
freq y;
run;
The data should have one column for block, treatment, lesion category, and
frequency or number of observations (Y ), which, in this case, is referenced by the
variables block, trt, category, and frequency, respectively.
Most of the options in the above syntax have already been explained previously;
the “order = data” option specifies that the order in which the categories appear in
the dataset will be treated as ordinal categories from the lowest to the highest for the
analysis. If this option is not used with the response variable in the model specification, “proc GLIMMIX” will rearrange its categories in an alphabetical or numerical order, but this will depend on whether the categories are entered as a number or a
name. The “freq y” option orders GLIMMIX to use y as the number of observations
in the corresponding category. The “estimate” command specifies the estimable
functions that form the boundaries between categories of each of the four treatments.
For example, the first estimate “c = 0, t = 1” defines η0 + τ1, that is, the boundary
between the categories “Without” (no lesion) and “Moderate” (slight lesion) for
treatment 1. This first estimate corresponds to logit log 1 -π 01π01 , which is the
probability that a chicken that received treatment 1 will respond to a degree of lesion
classified under category 0 (no lesion). The second estimation “c = 1, t = 1” defines
η1 + τ1, that is, the boundary between the categories “Moderate” (slight lesion) and
8.3
Cumulative Logit Models (Proportional Odds Models)
333
Table 8.9 Results of the analysis of variance in the multinomial cumulative logit model
(a) Type III tests of fixed effects
Effect
Num DF
Trt
3
(b) Solutions for fixed effects
Effect
Categoría
Without
Intercept ðη1 Þ
Moderate
Intercept ðη2 Þ
Trt ðτ1 Þ
Trt ðτ2 Þ
Trt ðτ3 Þ
Trt ðτ4 Þ
Den DF
794
Trt
1
2
3
4
Pr > F
<0.0001
F-value
22.45
Estimate
0.6144
3.8787
-1.5034
-0.2509
-1.0365
0
“Severe” (severe lesion) for treatment 1 and corresponds to logit log
Standard error
0.1799
0.2465
0.2086
0.2055
0.2036
.
π 11
1 - π 11
, and so
on. By taking the inverse of these links values, we can obtain the estimated
probabilities of π 01 and π 11. Part of the Statistical Analysis Software (SAS) glimmix
output is presented below:
The results of the analysis of variance in part (a) of Table 8.9 indicate that the
degree of lesion in the chicken footpad (pododermatitis) in the treatments tested were
significantly different (P < 0.0001). Therefore, the hypothesis of proportional odds
of treatments is rejected (H0 : τi = 0 for all i, that is, oddsratio = 1).
In part (b) of Table 8.9, we can see that the estimated intercepts η1 = 0:6144 and
η2 = 3:8787 define the boundary between the categories “Without” lesion
and “Moderate” lesion and the boundary between the categories “Moderate” lesion
and “Severe” lesion, respectively. The estimated effect of the treatments ðτi Þ shows
that the boundaries move either upward or downward when a certain treatment is
applied. In this sense, all estimated treatment coefficients have a negative effect with
respect to treatment 4. This means that chickens under treatments 1–3 have a low
probability of developing a moderate lesion and a higher probability of developing a
severe lesion than when treatment 4 is applied.
To calculate the probability that a chicken will not develop footpad dermatitis
(c = 0) when receiving treatment 1, that is, “c = 0, Trt = 1,” we first estimate the
linear predictor η01 = η0 þ τ1 = 0:6144 þ ð- 1:5034Þ = - 0:889, and, taking the
inverse, we obtain π 01 = 1=1þe - ð- 0:889Þ = 0:29. This value is the estimated probability
that a chicken will not develop footpad dermatitis when receiving treatment 1. However, now, for “c = 1, Trt = 1,” η11 = η1 þ τ1 = 3:8787 þ ð- 1:5034Þ = 2:3753,
whose inverse value is 0.915. This value is an estimate of the probability π 01 þ π 11 .
From this value, we obtain the probability that a chicken will develop a moderate
lesion and a severe lesion. For a moderate lesion, the probability is π 11 =
0:915 - π 01 = 0:915 - 0:29 = 0:624, and, for a severe lesion, the probability is
π 21 = 1 - 0:915 = 0:085. In a similar way the probabilities for the categories
(c = 0, 1, 2) of the rest of the treatments are computed.
334
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.10 Estimated odds ratio
Odds ratio estimates
Comparison
trt 1 vs. 4
trt 2 vs. 4
trt 3 vs. 4
Estimate
0.222
0.778
0.355
DF
794
794
794
95% Confidence limits
0.148
0.335
0.520
1.165
0.238
0.529
Table 8.11 Estimates on the model scale (Estimate) and on the data scale (Mean) for footpad
dermatitis categories in the multinomial cumulative logit model
Estimates
Label
c = 0,
t=1
c = 1,
t=1
c = 0,
t=2
c = 1,
t=2
c = 0,
t=3
c = 1,
t=3
c = 0,
t=4
c = 1,
t=4
Standard
error
0.2428
DF
794
t-value
-4.93
Pr > |t|
<0.0001
Mean
0.2914
Standard error
mean
0.001174
2.3753
0.2214
794
10.73
<0.0001
0.9149
0.01724
0.3634
0.1757
794
2.07
0.0390
0.5899
0.04252
3.6277
0.2420
794
14.99
<0.0001
0.9741
0.006103
-0.4222
0.1740
794
-2.43
0.0155
0.3960
0.04162
2.8422
0.2304
794
12.34
<0.0001
0.9449
0.01199
3.8787
0.2465
794
15.73
<0.0001
0.9797
0.004893
0.6144
0.1799
794
3.41
0.0007
0.6489
0.04098
Estimate
-0.8893
The odds ratios tabulated in Table 8.10 are the odds ratios for treatments 1 through
4, i.e., eτi for treatments 1–4. These are the estimated odds ratios of adjacent
categories of treatments i (i = 1, 2, 3) relative to treatment 4. Values of τi are not
category-specific; the odds ratios for “Without” lesion versus “Moderate” lesion and
those for “Moderate” lesion versus “Severe” lesion are listed below (hence the name
“proportional odds”).
From the above odds ratio results, it should be obvious why the F- and P-values
in the fixed effects tests are what they are. Adding the “ilink” option to the end of the
“estimate” command prompts GLIMMIX to estimate the inverse of the linear predictors ðηci Þ, i.e., the probabilities per category π ci = 1=1þe - ηci (Table 8.11).
In the above table, several estimates are shown for ηc þ τi . For example, the
probability that a chicken will not develop a lesion under treatment 1 can be
represented by “c = 0, t = 1,” that is, ηc þ τ1 = -0.8893. This result matches
the one obtained from the fixed effects table “Solutions for fixed effects”
previously shown. Taking the inverse of the link yields the probability
8.3
Cumulative Logit Models (Proportional Odds Models)
Without
0.680
Moderate
335
Severe
0.649
0.624
0.590
0.549
Probability of lesion
0.580
0.480
0.380
0.384
0.396
0.331
0.291
0.280
0.180
0.085
0.080
-0.020
0.055
0.026
Trt1
Trt2
Trt3
0.020
Trt4
Treatment
Fig. 8.1 Estimated probabilities for the footpad lesion categories in the treatments tested, using the
cumulative logit model
π 01 = 1=ð1 þ e0:8893 Þ = 0:2914. This probability is the maximum likelihood estimate
that a chicken will have no footpad lesion with treatment 1. The inverse of the link
function is under the “Mean” column of Table 8.11. Now, for the category “c = 1,
t = 1,” the inverse of the linear predictor is 0.9149, this is the estimate of π 01 þ π 11 .
From this value, we can obtain the probability of a chicken showing a “Moderate”
lesion when receiving treatment 1, that is, π 01 þ π 11 = 0:9149, and, substituting the
value of π 01 , we obtain the value π 11 = 0:9141 - 0:2914 = 0:6227. Finally, for a
“Severe” lesion (category“c = 2, t = 1”), the probability that a chicken will present a
severe lesion is π 21 = 1 - 0:9141 = 0:0859. Following the same procedure, we can
obtain the probabilities for each of the following categories (c = 0, 1, 2) of the rest of
the treatments (2–4).
Figure 8.1 shows that under the traditional feeding program with a litter density
of 1 kg m-2 of rice husks (Trt1), there is a high probability that broilers will
develop moderate and severe footpad lesions, as shown by π 11 = 0:624 and
π 21 = 0:085, respectively. When the litter density was increased from 1 to 2 kg m-2
of rice husks under the traditional broiler program (Trt2), the probability of the risk of
developing moderate and severe footpad lesions in broilers decreased significantly to
π 12 = 0:384 and π 22 = 0:026, respectively, compared to Trt1, whereas the probability
of not developing a footpad lesion increased to π 02 = 0:590 ðTrt2Þ compared to
π 01 = 0:291 ðTrt1Þ. Regarding the implementation of the two foot care programs
plus the litter density of 2 kg husk m-2 of rice husks, the probability of chickens of
not developing a footpad lesion is π 04 = 0:649 (Trt4) compared to π 03 = 0:396 in
Trt3, whereas the probability of chickens developing moderate and severe lesions
decreased from π 14 = 0:331 and π 24 = 0:025 in Trt4 compared to π 13 = 0:549 and
π 23 = 0:055 in Trt3.
336
8.4
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Cumulative Probit Models
An ordinal cumulative probit model, first considered by Aitchison and Silvey
(1957), generalizes a binary probit model to ordinal responses. This model results
from the probit modeling of the cumulative probabilities as a linear function of the
covariates. The link functions for the cumulative probit model with C categories are
listed below:
η1 = Φ - 1 ðπ 1 Þ = η1 þ Xβ þ Zb
η2 = Φ - 1 ðπ 1 þ π 2 Þ = η2 þ Xβ þ Zb
ηC - 1 = Φ
-1
⋮
ðπ 1 þ π 2 þ ⋯ þ π C - 1 Þ = ηC - 1 þ Xβ þ Zb
where X and Z are the design matrices, β and b are the vectors of fixed and random
effects parameters, respectively, and Φ-1() is the inverse function of the standard
normal cumulative distribution. The inverse link of each of the link functions is as
follows:
π 1 = Φðη1 Þ = hðη1 Þ
π 1 þ π 2 = Φðη2 Þ = hðη2 Þ
⋮
π 1 þ π 2 þ ⋯ þ π C - 1 = Φðηc - 1 Þ = hðηc - 1 Þ:
Once h(η1), h(η2), ... h(ηc - 1) are estimated, we can estimate π 1 , ... , π C . The
quality of the estimates of the ordinal cumulative probit model are usually very
similar to those of an ordinal cumulative logit model for some datasets but not all.
Both involve stochastic ordering at different levels of the response variable and are
designed to detect the location of changes in the response variable.
Returning to Example 8.3.1, for the cumulative probit model, we change the
“LINK = CPROBIT” option in the model’s definition of the above program syntax.
The output will contain all the same elements, except the odds ratios. The analysis
for the cumulative probit is exactly the same as that one we performed in the
cumulative logit model. Part of the output is shown in parts (a)–(c) of Table 8.12.
The estimated variance component due to blocks is σ 2block = 0:0092. The results of
the analysis of variance showed that the degrees of lesion in the chickens’ footpad
(pododermatitis) in the tested treatments differ significantly (P < 0.0001).
In part (b) of Table 8.12, it is possible to observe that the estimated intercepts
η1 = 0:3880 and η2 = 2:2407 define the boundary between the “Without” lesion and
“Moderate” lesion categories and the boundary between the “Moderate” lesion and
“Severe” lesion categories, respectively. The estimated effect of the treatments ðτi Þ
moves the boundaries either upward or downward, when a certain treatment is
applied. In this sense, all estimated treatment coefficients have a negative effect
8.4
Cumulative Probit Models
337
Table 8.12 Results of the analysis of variance in the multinomial cumulative probit model
(a) Covariance parameter estimates
Cov Parm
Subject
Intercept
Blk
(b) Type III tests of fixed effects
Effect
Num DF
Trt
3
(c) Solutions for fixed effects
Effect
Categoría
Without
Intercept ðη1 Þ
Moderate
Intercept ðη2 Þ
Trt ðτ1 Þ
Trt ðτ2 Þ
Trt ðτ3 Þ
Trt ðτ4 Þ
Estimate
0.009262
Den DF
794
Trt
1
2
3
4
Standard error
0.01817
F-value
24.57
Estimate
0.3880
2.2407
-0.9278
-0.1595
-0.6459
0
Pr > F
<0.0001
Standard error
0.1124
0.1375
0.1227
0.1242
0.1219
.
with respect to treatment 4. This means that chickens under treatments 1–3 have a
low probability of developing a footpad lesion and a higher probability of developing a severe lesion with respect to treatment 4.
From “Type III tests of fixed effects” (Table 8.12, part (b)), the probabilities for
each of the categories can be obtained. For the probability that a chicken will not
develop a footpad lesion (c = 0) under treatment 1, i.e., “c = 0, Trt = 1, ” the estimated
linear predictor is obtained as η01 = η0 þ τ1 = 0:3880 þ ð- 0:9278Þ = - 0:5398 and,
taking the inverse, gives π 01 = Φð- 0:5398Þ = 0:2946, that is, the estimated probability that a chicken will not develop a footpad lesion when receiving treatment 1. For
“
c = 1, Trt = 1, ” η11 = η1 þ τ1 = 2:2407 þ ð- 0:9278Þ = 1:3129, whose inverse
value is 0.9054. This value is an estimator of π 01 þ π 11 . From this value, we can
obtain the probability that a chicken will develop a moderate lesion and a severe
lesion. For a moderate lesion, π 11 = 0:9054 - π 01 = 0:9054 - 0:2946 = 0:6108, and,
for a severe lesion, π 21 = 1 - 0:9054 = 0:0946. Similarly, we can obtain the probabilities of the categories for the other treatments (c = 0, 1, 2) for the rest of the
treatments.
Similar to the previous example, adding the “ILINK” option to the end of the
“ESTIMATE” command prompts GLIMMIX to estimate the values of the linear
predictors ðηci Þ and the inverse of the linear predictors, which are the probabilities
per category ðπ ci = Φðηci ÞÞ. Table 8.13 shows the estimates of the linear predictors as
well as their inverse values (probabilities in this case).
From the above table, we show the estimates of ηc þ τi . For example, the estimated
linear predictor that a chicken will not develop a footpad lesion under treatment 1, i.e.,
“
c = 0, t = 1, ” is calculated as ηc þ τ1 = - 0:5398. This result matches the values
obtained from the fixed effects table (“Solutions for fixed effects”) previously shown.
Taking the inverse of the link function, π 01 = Φð0:5398Þ = 0:2947. This is the
338
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.13 Estimates on the model scale (Estimate) and on the data scale (Mean) for footpad
lesion categories in the multinomial cumulative probit model
Estimates
Label
c = 0,
t=1
c = 1,
t=1
c = 0,
t=2
c = 1,
t=2
c = 0,
t=3
c = 1,
t=3
c = 0,
t=4
c = 1,
t=4
Standard
error
0.1100
DF
794
tvalue
-4.91
Pr > |t|
<0.0001
Mean
0.2947
Standard error
mean
0.03793
1.3129
0.1208
794
10.87
<0.0001
0.9054
0.02035
0.2285
0.1105
794
2.07
0.0389
0.5904
0.04293
2.0812
0.1345
794
15.47
<0.0001
0.9813
0.006153
-0.2578
0.1085
794
-2.38
0.0178
0.3983
0.04189
1.5949
0.1258
794
12.68
<0.0001
0.9446
0.01407
2.2407
0.1375
794
16.29
<0.0001
0.9875
0.004457
0.3880
0.1124
794
3.45
0.0006
0.6510
0.04158
Estimate
-0.5398
probability that a chicken will not develop a footpad lesion when receiving treatment
1. This probability is under the “Mean” column.
Now, for the category “c = 1, t = 1, ” the inverse of the link function is a probability
of 0.9054, which results from the inverse value of the linear predictor η1 þ τ1 = 1:3129.
This value is the estimate in terms of probability π 01 þ π 11 . From this value, we can
obtain the probability that a chicken presents a “Moderate” lesion when receiving
treatment 1, that is, π 01 þ π 11 = 0:9054, and, using the value of π 01 , we obtain the
values π 11 = 0:9054 - 0:2947 = 0:6107 and π 21 = 1 - 0:9054 = 0:0946. Following
the same procedure, we can obtain the rest of the probabilities for each one of the
categories (c = 0, 1, 2) and for the rest of the treatments (2–4).
8.5
Effect of Judges’ Experience on Canned Bean Quality
Ratings
Canning quality is one of the most essential traits required in all new dry bean
(Phaseolus vulgaris L.) varieties, and the selection for this trait is a critical part of
bean breeding programs. Advanced lines that are candidates for release as varieties
must be evaluated for canning quality for at least 3 years from samples grown at
different locations. Quality is evaluated by a panel of judges with varying levels of
experience in evaluating breeding lines for visual quality traits. A total of 264 bean
breeding lines from 4 commercial classes were retained according to the procedures
described by Walters et al. (1997). These included 62 white (navy), 65 black,
8.5
Effect of Judges’ Experience on Canned Bean Quality Ratings
339
Table 8.14 Frequency of ratings of different types of beans as a function of the bean-rating
experience
Calif
1
2
3
4
5
6
7
Black
<
5 Years
13
91
123
72
24
2
0
>
5 Years
32
78
124
122
31
3
0
Kidney
<
5 Years
7
32
136
101
47
6
1
>
5 Years
10
31
96
104
71
18
0
Navy
<
5 Years
10
56
84
84
51
24
1
>
5 Years
22
51
107
98
52
37
5
Pinto
<
5 Years
13
29
91
109
60
25
1
>
5 Years
2
17
68
124
109
78
12
55 kidney, and 82 pinto bean lines plus control or “check” lines. The visual
appearance of the processed beans was determined subjectively by a panel of
13 judges on a 7-point hedonic scale (1 = very undesirable, ..., 4 = neither desirable
nor undesirable,..., 7 = very desirable). Beans were presented to the panel of judges
in random order at the same time. Before evaluating the samples, all judges were
shown examples of samples rated as satisfactory.
There is concern that certain judges, due to lack of experience, may not be able to
correctly score the canned samples. From attribute-based product evaluations, inferences about the effects of experience can be drawn from the psychology literature
(Wallsten and Budescu 1981). Prior to the bean canning quality rating experiment, it
was postulated that not only do less experienced judges have a more severe rating
than do more experienced judges but also that experience should have little or no
effect on white beans, for which the canning procedure was developed. Judges are
stratified for the purpose of analysis by experience (less than 5 years, greater than
5 years). Counts by canning quality, judge experience, and bean breeding lines are
listed in the following table (Table 8.14).
The link functions for the cumulative logit model for describing a variable with
C categories are as follows:
ηð1Þij = log
ηð2Þij = log
π 1ij
= η1 þ αi þ βj þ ðαβÞij
1 - π 1ij
π 1ij þ π 2ij
1 - π 1ij þ π 2ij
= η2 þ αi þ βj þ ðαβÞij
⋮
π 1ij þ π 2ij þ ⋯ þ π ðC - 1Þij
ηC - 1 = log
1 - π 1ij þ π 2ij þ ⋯ þ π ðC - 1Þij
= ηC - 1 þ αi þ βj þ ðαβÞij
The components of the GLMM with an ordinal multinomial response are as
follows:
340
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Distributions:
y1ij,
y2ij,
y3ij,
y4ij,
y5ij,
y6ij,y7ij~Multinomial
(Nij, π 1ij, π 2ij, π 3ij, π 4ij, π 5ij, π 6ij, π 7ij), where y1ij, y2ij, y3ij, y4ij, y5ij, y6ij, and y7ij are
the observed frequencies of the responses in each category c of the hedonic scale
(1 = very undesirable, ..., 4 = neither desirable nor undesirable, ..., 7 = very
desirable).
Linear predictor: η(c)ij = ηc + αi + βj + (αβ)ij, where η(c)ij is the cth link (c = 1, 2,...,6)
for bean type i and judge’s experience j; ηc is the intercept for the cth link; αi is the
fixed effect due to the bean type for ith bean class; βj is the fixed effect due to the
jth experience of the judge; and (αβ)ij is the fixed effect due to the interaction
between bean class and judge experience. The link functions for each category are
as follows:
log
log
log
log
log
log
π 1ij
= η1ij
1 - π 1ij
π 1ij þ π 2ij
1 - π 1ij þ π 2ij
= η2ij
π 1ij þ π 2ij þ π 3ij
1 - π 1ij þ π 2ij þ π 3ij
= η3ij
π 1ij þ π 2ij þ π 3ij þ π 4ij
1 - π 1ij þ π 2ij þ π 3ij þ π 4ij
= η4ij
π 1ij þ π 2ij þ π 3ij þ π 4ij þ π 5ij
1 - π 1ij þ π 2ij þ π 3ij þ π 4ij þ π 5ij
π 1ij þ π 2ij þ π 3ij þ π 4ij þ π 5ij þ π 6ij
1 - π 1ij þ π 2ij þ π 3ij þ π 4ij þ π 5ij þ π 6ij
= η5ij
= η6ij
The following GLIMMIX commands fit a cumulative logit model with an ordinal
multinomial response.
proc glimmix data=beans ;
class Exper;
model cal(order=data)= Exper|Class/dist=Multinomial link=clogit
8.5
Effect of Judges’ Experience on Canned Bean Quality Ratings
341
solution oddsratio;
Contrast 'Effect of Experience on Black bean' exper 1 -1 class*exper 1 -1
0 0 0 0 0 0 0 0 0 0;
Contrast 'Effect of Experience on Kidney Bean' exper 1 -1 class*exper 0 0
1 -1 0 0 0 0 0 0 0 0;
Contrast 'Effect of Experience on Navies bean' exper 1 -1 class*exper 0 0
0 0 0 0 0 1 -1 0 0 0;
Contrast 'Effect of Experience on Pinto beans' exper 1 -1 class*exper 0 0
0 0 0 0 0 0 0 0 0 1 -1;
estimate 'Black, < 5 year, Rating = 1' Intercept 1 0 0 0 0 0 0 0 0 class 1 0
0 0 0 0 0 exper 1 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, < 5 year, Rating <= 2' Intercept 0 1 0 0 0 0 0 0 0 class 1 0
0 0 0 0 0 exper 1 0 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, < 5 year, Rating <= 3' Intercept 0 0 0 1 0 0 0 0 0 class 1 0
0 0 0 0 0 exper 1 0 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, < 5 year, Rating <= 4' Intercept 0 0 0 0 1 0 0 0 class 1 0 0 0
0 0 0 exper 1 0 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, < 5 year, Rating <= 5' Intercept 0 0 0 0 0 0 1 0 0 class 1 0
0 0 0 0 0 exper 1 0 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, > 5 year, Rating <= 6' Intercept 0 0 0 0 0 0 0 0 1 class 1 0
0 0 0 0 exper 1 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, > 5 year, Rating = 1' Intercept 1 0 0 0 0 0 0 0 0 class 1 0
0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, > 5 year, Rating <= 2' Intercept 0 1 0 0 0 0 0 0 0 class 1 0
0 0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, > 5 year, Rating <= 3' Intercept 0 0 0 1 0 0 0 0 0 class 1 0
0 0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, > 5 year, Rating <= 4' Intercept 0 0 0 0 1 0 0 0 class 1 0 0 0
0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, > 5 year, Rating <= 5' Intercept 0 0 0 0 0 0 1 0 0 class 1 0
0 0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0 0/ilink;
estimate 'Black, > 5 year, Rating <= 6' Intercept 0 0 0 0 0 0 0 0 1 class 1 0
0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0 0/ilink;
estimate 'Kidney, < 5 year, Rating = 1' Intercept 1 0 0 0 0 0 0 0 0 class 0 1
0 0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink;
estimate 'Kidney, < 5 year, Rating <= 2' Intercept 0 1 0 0 0 0 0 0 0 class 0 1
0 0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink;
estimate 'Kidney, < 5 yr, Rating <= 3' Intercept 0 0 0 1 0 0 0 0 0 class 0 1
0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink;
estimate 'Kidney, < 5 year, Rating <= 4' Intercept 0 0 0 0 0 1 0 0 0 class 0 1
0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink;
estimate 'Kidney, < 5 year, Rating <= 5' Intercept 0 0 0 0 0 0 1 0 0 class 0 1
0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink;
estimate 'Kidney, < 5 year, Rating <= 6' Intercept 0 0 0 0 0 0 0 0 1 class 0 1
0 0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink;
estimate 'Kidney, > 5 year, Rating = 1' Intercept 1 0 0 0 0 0 0 0 0 class 0 1
0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink;
estimate 'Kidney, > 5 year, Rating <= 2' Intercept 0 1 0 0 0 0 0 0 0 class 0 1
0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink;
estimate 'Kidney, > 5 year, Rating <= 3' Intercept 0 0 0 1 0 0 0 0 0 class 0 1
0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink;
estimate 'Kidney, > 5 year, Rating <= 4' Intercept 0 0 0 0 0 1 0 0 0 class 0 1
0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink;
342
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
estimate 'Kidney, > 5 year, Rating <= 5' Intercept 0 0 0 0 0 0 1 0 0 class 0 1
0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink;
estimate 'Kidney, > 5 year, Rating <= 6' Intercept 0 0 0 0 0 0 0 0 1 class 0 1
0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink;
estimate 'Navies, < 5 year, Rating = 1' Intercept 1 0 0 0 0 0 0 0 0 class 0 0
0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink;
estimate 'Navies, < 5 year, Qualification <= 2' Intercept 0 1 0 0 0 0 0 0
0 class 0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink;
estimate 'Navies, < 5 year, Qualification <= 3' Intercept 0 0 0 0 1 0 0 0 0 0
class 0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink;
estimate 'Navies, < 5 year, Rating <= 4' Intercept 0 0 0 0 0 1 0 0 0 class 0 0
0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink;
estimate 'Navies, < 5 year, Rating <= 5' Intercept 0 0 0 0 0 0 0 1 0 0 class
0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink;
estimate 'Navies, <5 year, Qualification <= 6' Intercept 0 0 0 0 0 0 0 0 0 1
class 0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink;
estimate 'Navies, > 5 year, Qualification = 1' Intercept 1 0 0 0 0 0 0 0
0 class 0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink;
estimate 'Navies, > 5 year, Qualification <= 2' Intercept 0 1 0 0 0 0 0 0
0 class 0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink;
estimate 'Navies, > 5 year, Rating <= 3' Intercept 0 0 0 1 0 0 0 0 0 class 0 0
0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink;
estimate 'Navies, > 5 year, Rating <= 4' Intercept 0 0 0 0 0 1 0 0 0 class 0 0
0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink;
estimate 'Navies, > 5 year, Rating <= 5' Intercept 0 0 0 0 0 0 0 1 0 0 class
0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink;
estimate 'Navies, > 5 year, Rating <= 6' Intercept 0 0 0 0 0 0 0 0 0 1 class
0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink;
estimate 'Pinto, < 5 year, Qualification = 1' Intercept 1 0 0 0 0 0 0 0
0 class 0 0 0 0 0 0 1 exper 1 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink;
estimate 'Pinto, < 5 year, Qualification <= 2' Intercept 0 1 0 0 0 0 0 0
0 class 0 0 0 0 0 0 1 exper 1 0 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink;
estimate 'Pinto, < 5 year, Qualification <= 3' Intercept 0 0 0 0 1 0 0 0 0 0
class 0 0 0 0 0 1 exper 1 0 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink;
estimate 'Pinto, < 5 year, Rating <= 4' Intercept 0 0 0 0 0 1 0 0 0 class 0 0
0 0 0 1 exper 1 0 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink;
estimate 'Pinto, < 5 year, Rating <= 5' Intercept 0 0 0 0 0 0 0 1 0 0 class 0 0
0 0 0 1 exper 1 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink;
estimate 'Pinto, < 5 year, Rating <= 6' Intercept 0 0 0 0 0 0 0 0 0 1 class 0 0
0 0 0 1 exper 1 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink;
estimate 'Pinto, > 5 years, Qualification = 1' Intercept 1 0 0 0 0 0 0 0
0 class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 1/ilink;
estimate 'Pinto, > 5 year, Qualification <= 2' Intercept 0 1 0 0 0 0 0 0
0 class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 0 1/ilink;
estimate 'Pinto, > 5 year, Qualification <= 3' Intercept 0 0 0 1 0 0 0 0
0 class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 0 1/ilink;
estimate 'Pinto, > 5 year, Rating <= 4' Intercept 0 0 0 0 0 1 0 0 0 class 0 0
0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 0 1/ilink;
estimate 'Pinto, > 5 year, Rating <= 5' Intercept 0 0 0 0 0 0 0 1 0 0 class 0 0
0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 0 1/ilink;
estimate 'Pinto, > 5 year, Qualification <= 6' Intercept 0 0 0 0 0 0 0 0 0 1
class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 1/ilink;
freq y;
run;
8.5
Effect of Judges’ Experience on Canned Bean Quality Ratings
Table 8.15 Fixed effects
hypothesis testing in the multinomial cumulative logit
model
Type III tests of fixed effects
Effect
Num DF
Class
1
Exper
3
Class*Exper
3
Den DF
2779
2779
2779
343
F-value
36.19
85.20
10.13
Pr > F
<0.0001
<0.0001
<0.0001
Table 8.16 Hypothesis testing in quality assessment
Contrasts
Label
Effect of experience on black beans
Effect of experience on kidney beans
Effect of experience on navy beans
Effect of experience on pinto beans
Num DF
1
1
1
1
Den DF
2779
2779
2779
2779
F-value
2.77
7.86
0.02
58.06
Pr > F
0.0961
0.0051
0.8822
<0.0001
Part of the results is shown below. The results of the analysis of variance show
that the class of bean (Class), experience of the evaluator (Exper), and the interaction
between class and experience (Class×Exper) on bean canning scores differ significantly (P = 0.0001). That is, the results of comparing judges with more and less
years of experience will depend on the line (variety) of beans (Table 8.15).
The contrasts address this interaction (Table 8.16). Hypothesis testing is as
follows: π class of bean, < 5 years of experience = π class of bean, > 5 years of experience.
The results show that judges with more than 5 years of experience differ from
those with less than 5 years of experience in evaluating the quality of canned kidney
and pinto beans (Table 8.16). With the “solution” option in the model specification,
the fixed parameter estimates table shows the solution of the fixed effects parameters
under maximum likelihood. In this table, we can observe the values of the estimated
intercepts: η1 = - 4:6421 defines the boundary between the categories, “1 = highly
undesirable” and “2 = moderately undesirable”, whereas η2 = - 2:9316 defines the
boundary between the categories “2 = moderately undesirable” and “3 = slightly
undesirable.” The third intercept defines the boundary between the categories
“3 = moderately undesirable” and “3 = slightly undesirable,” η3 = - 1:3995 defines
the boundary between the categories “3 = slightly undesirable” and “4 = neither
undesirable nor desirable,” and so on.
The estimated effects of bean type ðαi Þ, evaluator βi , and their interaction αβij are
shown below. From these values, we can estimate the linear predictors for each of the
categories. For example, the linear predictor for canned black beans evaluated by an inexperienced judge who assigns the category “1 = very undesirable” is η111 = η1 þ α1 þ β1 þ
αβ11 = - 4:6421 þ 1:9670 þ 1:0284 - 0:8066 = - 2:4533, for category “2 = moderately undesirable,” it is η211 = η2 þ α1 þ β1 þ αβ11 = - 2:9316 þ 1:9670 þ
1:0284 - 0:8066 = - 0:7428,
for category “3 = slightly undesirable,” it is
η311 = η3 þ α1 þ β1 þ αβ11 = - 1:3995 þ 1:9670 þ 1:0284 - 0:8066 = 0:7893, and,
344
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.17 Maximum likelihood estimation of the estimated parameters in the fixed effects
solution of canned bean quality ratings in the multinomial cumulative logit model
Fixed parameter estimates
Effect
Intercept η1
Intercept η2
Intercept η3
Intercept η4
Intercept η5
Intercept η6
Class
Class
Class
Class
Cal
ηi
1
2
3
4
5
6
Class
αi
Expert
β1
Black
α1
Kidney
α2
Navy
α3
Pinto
α4
1 β1
2
Exper
Exper
Class*Exper
Black
Class*Exper
Class*Exper
Black
Kidney
Class*Exper
Class*Exper
Kidney
Navy
Class*Exper
Class*Exper
Navy
Pinto
Class*Exper
Pinto
1 αβ11
2
1 αβ21
2
1 αβ31
2
1 αβ41
2
Estimate
-4.6421
-2.9316
-1.3995
0.004287
1.4191
3.8925
1.9670
Standard
error
0.1363
0.1057
0.09643
0.09230
0.1026
0.2346
0.1318
DF
2779
2779
2779
2779
2779
2779
2779
t-value
-34.05
-27.74
-14.51
0.05
13.84
16.59
14.93
Pr > |t|
<0.0001
<0.0001
<0.0001
0.9630
<0.0001
<0.0001
<0.0001
1.0472
0.1342
2779
7.80
<0.0001
1.3076
0.1345
2779
9.72
<0.0001
0
.
.
1.0284
0.1350
2779
7.62
<0.0001
0
-0.8066
.
0.1894
.
2779
.
-4.26
.
<0.0001
0
-0.6457
.
0.1912
.
2779
.
-3.38
.
0.0007
0
-1.0072
.
0.1969
.
2779
.
-5.12
.
<0.0001
0
0
.
.
.
.
.
.
.
.
0
.
.
.
.
.
.
for
category
“4
=
neither
undesirable
nor
desirable,”
it
is
η411 = η4 þ α1 þ β1 þ αβ11 = 0:004287 þ 1:9670 þ 1:0284 - 0:8066 = 2:1931. This
is how the other categories are calculated for each type of bean and assessor (Table 8.17).
The results of Table 8.18 were obtained with the “estimate” command in conjunction with the “ilink” option that prompts GLIMMIX to compute the values of the
linear predictors, ^ηcij , tabulated under the “Estimate” column, and the estimated
probabilities π^cij for all categories of each treatment are tabulated under the “Mean”
column π cij , except the reference category.
From Table 8.18 (“Estimates”), we can obtain the probabilities reported under the
“Mean” column in which an inexperienced (<5 years) panelist (judge) would rate
canned black beans as category 1 (1 = highly undesirable) with a probability of
π^111 = 0:08 compared to an experienced panelist (>5 years) who would give a
8.5
Effect of Judges’ Experience on Canned Bean Quality Ratings
345
Table 8.18 Estimates on the model scale (Estimate) and on the data scale (Mean) based on judges’
experience in canned bean quality ratings in the multinomial cumulative logit model
Estimates
Label
Black
<5 years,
score = 1
Black
<5 years,
score ≤ 2
Black
<5 years,
score ≤ 3
Black
<5 years,
score ≤ 4
Black
<5 years,
score ≤ 5
Black
>5 years,
score ≤ 6
Black
>5 years,
score = 1
Black
>5 years,
score ≤ 2
Black
>5 years,
score ≤ 3
Black
>5 years,
score ≤ 4
Black
>5 years,
score ≤ 5
Black
>5 years,
score ≤ 6
Kidney
<5 years,
score = 1
Kidney
<5 years,
score ≤ 2
Estimate
-2.4533
Standard
error
0.1292
DF
2779
t-value
-18.99
Pr > |t|
<0.0001
Mean
0.07920
Standard
error
mean
0.009419
-0.7428
0.1004
2779
-7.40
<0.0001
0.3224
0.02194
0.7893
0.1008
2779
7.83
<0.0001
0.6877
0.02164
2.1931
0.1076
2779
20.38
<0.0001
0.8996
0.009716
3.6079
0.1238
2779
29.15
<0.0001
0.9736
0.003180
6.0814
0.2467
2779
24.65
<0.0001
0.9977
0.000561
-2.6751
0.1264
2779
-21.17
<0.0001
0.06446
0.007621
-0.9646
0.09577
2779
-10.07
<0.0001
0.2760
0.01913
0.5675
0.09314
2779
6.09
<0.0001
0.6382
0.02151
1.9713
0.09967
2779
19.78
<0.0001
0.8778
0.01069
3.3861
0.1170
2779
28.95
<0.0001
0.9673
0.003704
5.8595
0.2434
2779
24.07
<0.0001
0.9972
0.000690
-3.2122
0.1333
2779
-24.11
<0.0001
0.03871
0.004958
-1.5017
0.1018
2779
-14.74
<0.0001
0.1822
0.01517
(continued)
346
8 Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.18 (continued)
Estimates
Label
Kidney
<5 years,
score ≤ 3
Kidney
<5 years,
score ≤ 4
Kidney
<5 years,
score ≤ 5
Kidney
>5 years,
score ≤ 6
Kidney
>5 years,
score = 1
Kidney
>5 years,
score ≤ 2
Kidney
>5 years,
score ≤ 3
Kidney
>5 years,
score ≤ 4
Kidney
>5 years,
score ≤ 5
Kidney
>5 years,
score ≤ 6
Navies
<5 years,
score = 1
Navies
<5 years,
score ≤ 2
Navies
<5 years,
score ≤ 3
Navies
<5 years,
score ≤ 4
Navies
<5 years,
score ≤ 5
Standard
error
0.09608
DF
2779
t-value
0.32
Pr > |t|
0.7517
Mean
0.5076
Standard
error
mean
0.02401
1.4342
0.1011
2779
14.19
<0.0001
0.8076
0.01571
2.8490
0.1178
2779
24.18
<0.0001
0.9453
0.006096
5.3225
0.2438
2779
21.83
<0.0001
0.9951
0.001179
-3.5949
0.1372
2779
-26.20
<0.0001
0.02673
0.003569
-1.8844
0.1071
2779
-17.60
<0.0001
0.1319
0.01226
-0.3523
0.09988
2779
-3.53
0.0004
0.4128
0.02421
1.0515
0.1020
2779
10.31
<0.0001
0.7411
0.01957
2.4663
0.1176
2779
20.98
<0.0001
0.9217
0.008480
4.9397
0.2436
2779
20.27
<0.0001
0.9929
0.001719
-3.3133
0.1404
2779
-23.60
<0.0001
0.03512
0.004757
-1.6027
0.1119
2779
-14.33
<0.0001
0.1676
0.01561
-0.07066
0.1068
2779
-0.66
0.5084
0.4823
0.02667
1.3332
0.1102
2779
12.10
<0.0001
0.7914
0.01820
2.7479
0.1251
2779
21.97
<0.0001
0.9398
0.007077
Estimate
0.03040
(continued)
8.5
Effect of Judges’ Experience on Canned Bean Quality Ratings
347
Table 8.18 (continued)
Estimates
Label
Navies
>5 years,
score ≤ 6
Navies
>5 years,
score = 1
Navies
>5 years,
score ≤ 2
Navies
>5 years,
score ≤ 3
Navies
>5 years,
score ≤ 4
Navies
>5 years,
score ≤ 5
Navies
>5 years,
score ≤ 6
Pinto
<5 years,
score = 1
Pinto
<5 years,
score ≤ 2
Pinto
<5 years,
score ≤ 3
Pinto
<5 years,
score ≤ 4
Pinto
<5 years,
score ≤ 5
Pinto
>5 years,
score ≤ 6
Pinto
>5 years,
score = 1
Pinto
>5 years,
score ≤ 2
Estimate
5.2214
Standard
error
0.2473
DF
2779
t-value
21.12
Pr > |t|
<0.0001
Mean
0.9946
Standard
error
mean
0.001321
-3.3345
0.1348
2779
-24.74
<0.0001
0.03441
0.004479
-1.6240
0.1047
2779
-15.51
<0.0001
0.1647
0.01440
-0.09190
0.09897
2779
-0.93
0.3532
0.4770
0.02469
1.3119
0.1028
2779
12.76
<0.0001
0.7878
0.01719
2.7267
0.1186
2779
22.99
<0.0001
0.9386
0.006836
5.2002
0.2439
2779
21.32
<0.0001
0.9945
0.001331
-3.6137
0.1380
2779
-26.19
<0.0001
0.02624
0.003527
-1.9032
0.1081
2779
-17.61
<0.0001
0.1297
0.01221
-0.3711
0.1008
2779
-3.68
0.0002
0.4083
0.02436
1.0327
0.1030
2779
10.03
<0.0001
0.7374
0.01994
2.4475
0.1184
2779
20.67
<0.0001
0.9204
0.008678
4.9210
0.2439
2779
20.17
<0.0001
0.9928
0.001753
-4.6421
0.1363
2779
-34.05
<0.0001
0.009545
0.001289
-2.9316
0.1057
2779
-27.74
<0.0001
0.05061
0.005078
(continued)
348
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.18 (continued)
Estimates
Label
Pinto
>5 years,
score ≤ 3
Pinto
>5 years,
score ≤ 4
Pinto
>5 years,
score ≤ 5
Pinto
>5 years,
score ≤ 6
Standard
error
0.09643
DF
2779
t-value
-14.51
Pr > |t|
<0.0001
Mean
0.1979
Standard
error
mean
0.01531
0.004287
0.09230
2779
0.05
0.9630
0.5011
0.02307
1.4191
0.1026
2779
13.84
<0.0001
0.8052
0.01609
3.8925
0.2346
2779
16.59
<0.0001
0.9800
0.004595
Estimate
-1.3995
probability of π^112 = 0:0646. To calculate the probability that a judge with less than
5 years experience would assign a rating of 2 (2 = moderately undesirable) to canned
black beans, we derive this probability from the cumulative probability of 0.3224,
from
which
we
get
which
corresponds
to
π 211 þ π 111 ,
π 211 = 0:3224 - π 111 = 0:3224 - 0:08 = 0:24. On the other hand, for a judge with
experience (>5 years), the probability of assigning a score of 2 to canned black
beans is π 212 = 0:2760 - π 112 = 0:2760 - 0:06446 = 0:2115.
Following the same procedure, the other probabilities for the rest of the categories
are obtained. The probabilities calculated for each of the categories are shown in
Table 8.19 and can be seen in Fig. 8.2.
8.6
Generalized Logit Models: Nominal Response Variables
In a model with unordered data, the polytomous response variable does not have an
ordered structure. Two classes of models, generalized logit models and conditional
logit models, can be used with nominal response data. A generalized logit model
consists of a combination of several binary logits estimated simultaneously. A logit
model is the simplest and best-known probabilistic choice model. However, there are
problems in making use of a multinomial logit model because of its inflexibility. A
generalized logit model is essentially more flexible than the traditional multinomial
cumulative logit model.
A generalized logit model shows the same flexibility as a probit model but is
much more tractable. Like cumulative logit and probit models, a generalized logit
model has C – 1 link functions, where C denotes the number of response categories.
8.6
Generalized Logit Models: Nominal Response Variables
349
Table 8.19 Probabilities calculated for each of the canned bean grades
Black
Kidney
Navy
Pinto
Cal1
0.08
0.06
0.04
0.03
0.04
0.03
0.03
0.01
J1
J2
J1
J2
J1
J2
J1
J2
Cal2
0.24
0.21
0.14
0.11
0.13
0.13
0.10
0.04
Cal3
0.37
0.36
0.33
0.28
0.31
0.31
0.28
0.15
Cal4
0.21
0.24
0.30
0.33
0.31
0.31
0.33
0.30
Cal5
0.07
0.09
0.14
0.18
0.15
0.15
0.18
0.30
Cal6
0.02
0.03
0.05
0.07
0.05
0.06
0.07
0.17
Cal7
0.00
0.00
0.00
0.01
0.01
0.01
0.01
0.02
Cal1 = qualification 1, Cal2 = qualification 2,...., Cal7 = qualification 7; J1 = panelist with less
than 5 years’ experience, and J2 = panelist with more than 5 years’ experience
0.17
0.33
0.30
0.18
0.15
0.15
0.31
0.18
0.33
0.02
0.01
0.07
J1
Black
Cal1
J1
Kidney
Cal2
Cal3
J2
Cal5
Cal6
0.15
0.04
0.03
0.03 0.01
J1
Navy
Cal4
0.30
0.31
0.31
J2
0.04
0.28
J2
0.03
0.10
J1
0.04
0.13
0.06
0.13
0.08
0.11
0.0
0.14
0.1
0.21
0.3
0.28
0.33
0.4
0.2
0.01
0.06
0.36
0.5
0.37
0.6
0.01
0.05
0.01
0.07
0.31
0.00
0.03 0.05
0.14
0.00
0.30
0.7
0.02
0.24 0.09
0.8
0.21 0.07
0.9
0.00
0.24
Probability for each category
1.0
J2
Pinto
Cal7
Fig. 8.2 Estimated probabilities for each category of the acceptability of canned beans, according
to the experience of the panelist (judge)
Moreover, in this class of models, a category is first defined as the reference
category. This may be arbitrary or it may make compelling logical sense in the
study to designate a particular response category as the reference. In practice and
throughout the analysis, the category used as the reference is irrelevant, as long as we
are consistent about it. For example, if C is used as the reference category, then the
generalized logits are defined as shown below:
350
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
η1 = log
π 1ij
= α1 þ Xβ1 þ Zb1
π cij
η2 = log
π 2ij
= α2 þ Xβ2 þ Zb2
π Cij
⋮
π ðC - 1Þij
ηC - 1 = log
= αc - 1 þ XβC - 1 þ ZbC - 1
π Cij
Given the different effects in the models, the intercepts (α´s), β´s, and b´s vary
across the pairs of response variable categories for each link function. Using algebra,
it can be shown that the general form of the inverse of the link functions is given by
eη c
πc =
1þ
Once π 1, π 1, . . . . π C
πC = 1 -
8.6.1
C-1
c=1
- 1
C-1
,
c = 1, 2, . . . , C - 1
eη c
c=1
are estimated, the reference category is estimated as
πc.
CRDs with a Nominal Multinomial Response
In practice, cumulative models are used for analyzing ordinal data and generalized
logit models for nominal data. Returning to Example 8.3.1, we will now implement
the analysis of a generalized logit model. This model relaxes the assumptions of
proportionality; but it is less parsimonious than the “odds ratio” model since they fit
C - 1 binary logit models, where C is the number of categories of the response
variable. The linear predictor and distribution are the same as in the previous
example.
The following GLIMMIX syntax implements the analysis of the generalized logit
model:
proc glimmix data=chickens ;
class trt block category;
model category(reference='severe')= trt/dist=Multinomial
link=glogit oddsratio;
random intercept/subject=block solution group=category;
estimate 't=1' intercept 1 trt 1 0,
t=2' intercept 1 trt 0 1 0,
't=3' intercept 1 trt 0 0 0 1 0,
't=4' intercept 1 trt 0 0 0 0 1/ilink bycat;
freq y;
run;
8.6
Generalized Logit Models: Nominal Response Variables
Table 8.20 Analysis of
variance in the generalized
multinomial logit model
Type III tests of fixed effects
Effect
Num DF
Den DF
Trt
6
790
351
F-value
10.78
Pr > F
<0.0001
Table 8.21 Maximum likelihood estimates on the model scale (Estimate) for footpad lesion level
in the multinomial generalized logit model
Solutions for fixed effects
Effect
Category
Intercept
Without lesion
Intercept
Moderate lesion
trt
Without lesion
trt
Moderate lesion
trt
Without lesion
trt
Moderate lesion
trt
Without lesion
trt
Moderate lesion
trt
Without lesion
trt
Moderate lesion
Trt
1
1
2
2
3
3
4
4
Estimate
4.8525
4.2485
-3.8447
-2.6478
-1.1888
-0.9651
-2.7860
-1.8326
0
0
Standard error
1.0059
1.0071
1.0330
1.0327
1.1618
1.1662
1.0585
1.0598
.
.
DF
2
2
790
790
790
790
790
790
.
.
t-value
4.82
4.22
-3.72
-2.56
-1.02
-0.83
-2.63
-1.73
.
.
Pr > |t|
0.0404
0.0519
0.0002
0.0105
0.3065
0.4082
0.0087
0.0842
.
.
Most of the syntax of the program has already been explained. The “reference=”
option is new to this program in the command, where the model is defined and is
used to designate the reference category. By not specifying the “reference=” option,
GLIMMIX by default uses the last category in the dataset. Moreover, the
“link = glogit” option prompts GLIMMIX to fit a generalized logit model. The
“bycat” option in the “estimate” command is unique to the generalized logit model.
Finally, the “ilink” option asks GLIMMIX to estimate all category probabilities for
each treatment, except those for the reference category. Part of the output is shown in
Table 8.20. The fixed effects test shows that there are highly significant differences
(P = 0.0001) on the average percentage of footpad lesion level between treatments.
Unlike the cumulative logit model, in the generalized logit model, the estimates
of the fixed effects (treatments), as well as the intercepts, are separated for each
link function. For the estimation of linear predictors, we use the estimated values
of Table 8.21 (“Solutions for fixed effects”). The estimated intercepts α1 = 4:8525 and
α2 = 4:2485 define the boundary between the categories “Without” lesion and “Moderate” lesion and the boundary between the categories “Moderate” lesion and “Severe”
lesion, respectively. For treatment 1, the treatment effects (^τi Þ estimated for the
“Without” lesion category is τ1 = - 3:8447 and for the “Moderate” lesion category,
it is τ1 = - 2:6478. With these values, the linear predictors for the “Without” lesion
and “Moderate” lesion categories under treatment 1 are η01 = 4:8525 - 3:8447 =
1:0077 and η11 = 4:2485 - 2:6478 = 1:6007, respectively.
The estimated probabilities for each of the categories (“Without” lesion
and “Moderate” lesion) in each treatment, except for the reference category, are found
352
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.22 Estimates on the model scale (“Estimate”) and on the data scale (“Mean”) for footpad
lesion level observed in treatments in the multinomial generalized logit model
Estimates
Label
t=1
t=1
t=2
t=2
t=3
t=3
t=4
t=4
Category
Without
lesion
Moderate
lesion
Without
lesion
Moderate
lesion
Without
lesion
Moderate
lesion
Without
lesion
Moderate
lesion
Estimate
1.0077
Standard
error
0.2515
DF
790
t-value
4.01
Pr > |t|
<0.0001
Mean
0.3150
Standard
error mean
0.03552
1.6007
0.2286
790
7.00
<0.0001
0.5700
0.03677
3.6637
0.5881
790
6.23
<0.0001
0.5850
0.03801
3.2834
0.5881
790
5.58
<0.0001
0.4000
0.03761
2.0665
0.3414
790
6.05
<0.0001
0.3929
0.03755
2.4159
0.3300
790
7.32
<0.0001
0.5573
0.03762
4.8525
1.0059
790
4.82
<0.0001
0.6433
0.03687
4.2485
1.0071
790
4.22
<0.0001
0.3517
0.03669
under the “Mean” column of Table 8.22. The probability that a chick has no footpad
lesion when receiving treatment 1 is π 01 = 0:315, whereas the value 0.57 corresponds to
the cumulative probability π 01 þ π 11 . From this value, we can calculate the probability
of observing a moderate lesion, which is π 11 = 0:57 - π 01 = 0:57 - 0:315 = 0:255.
From these probabilities, we can estimate the probability of observing a severe footpad
lesion under treatment 1 as π 21 = 1 - ð0:57Þ = 0:43. Following the same logic, we can
estimate the reference probabilities for the rest of the other treatments.
Another important result is the odds ratio estimates. These estimates are shown in
Table 8.23.
These odds ratios compare the odds for the labeled category to those for the
reference category for treatments 1–3 relative to treatment 4. These odds ratio values
are derived from the estimated probabilities in each of the categories. For example,
the probabilities that a chicken does not present a lesion and a moderate lesion
are π 04 = 0:6433 and π 14 = 0:3517, respectively. From these probabilities, we
can estimate the probability of observing a severe lesion as follows:
π 24 = 1 - ð0:6433 þ 0:3517Þ = 0:005. The estimated odds ratio of not observing a
lesion (“Without” lesion) between treatments 1 and 4 is
Odds ratioTrt1,Trt4 =
π 01 π 04 0:315 0:6433
=
=
=
= 0:0213
π 21 π 24 0:115 0:005
the value provided in the odds ratio estimates table. If we compare the analysis using
the cumulative logit link and the generalized logit link, we observe insignificant
8.6
Generalized Logit Models: Nominal Response Variables
353
Table 8.23 Estimated odds ratio
Odds ratio estimates
Category
Without lesion
Moderate lesion
Without lesion
Moderate lesion
Without lesion
Moderate lesion
Trt
1
1
2
2
3
3
_trt
4
4
4
4
4
4
Estimate
0.021
0.071
0.305
0.381
0.062
0.160
DF
790
790
790
790
790
790
95% Confidence limits
0.003
0.163
0.009
0.538
0.031
2.979
0.039
3.759
0.008
0.493
0.020
1.281
changes in the estimated category probabilities by treatment as well as in the
significance level in the test of treatment effects.
8.6.2
CRD: Cheese Tasting
Consider a study in which you want to know the effects of various additives on the
flavor of cheese. Researchers tested 4 cheese additives and obtained 52 response
ratings for each additive. Each response was measured on a scale of 9 categories
ranging from: I dislike it very much (1) to I like it very much or excellent flavor (9).
Data are obtained from the study by McCullagh and Nelder (1989) (Table 8.24).
The components of the GLMM with an ordinal multinomial response are as
follows:
Distributions: y1i, y2i, y3i, y4i, y5i, y6i,y7i, y8i, y9i~Multinomial
(Ni, π 1i, π 2i, π 3i, π 4i, π 5i, π 6i, π 7i,π 8i, π 9i), where y1i, y2i, y3i, y4i, y5i, y6i,y7i, y8i,
and y9i are the observed frequencies of the responses in each category c of the
hedonic scale (1 = very undesirable, ..., 5 = neither desirable nor undesirable, ... ,
9 = very desirable).
Linear predictor: η(c)i = ηc + αi, where η(c)ij is cth link (c = 1, 2, . . ., 8) for the additive
type i, ηc is the intercept for the cth link, and αi is the fixed effect due to the ith
additive. The link functions for each category are as follows:
log
log
log
π 1i
= η1i
1 - π 1i
π 1i þ π 2i
= η2i
1 - ðπ 1i þ π 2i Þ
π 1i þ π 2i þ π 3i
= η3i
1 - ðπ 1i þ π 2i þ π 3i Þ
354
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.24 Effect of additives on cheese flavor
id
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
log
log
log
log
log
Additive
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
Y
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
Freq
0
0
1
7
8
8
19
8
1
6
9
12
11
7
6
1
0
0
Id
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
Additive
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
Y
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
Freq
1
1
6
8
23
7
5
1
0
0
0
0
1
3
7
14
16
11
π 1i þ π 2i þ π 3i þ π 4i
= η4i
1 - ðπ 1i þ π 2i þ π 3i þ π 4i Þ
π 1i þ π 2i þ π 3i þ π 4i þ π 5i
= η5i
1 - ðπ 1i þ π 2i þ π 3i þ π 4i þ π 5i Þ
π 1i þ π 2i þ π 3i þ π 4i þ π 5i þ π 6i
= η6i
1 - ðπ 1i þ π 2i þ π 3i þ π 4i þ π 5i þ π 6i Þ
π 1i þ π 2i þ π 3i þ π 4i þ π 5i þ π 6i þ π 7i
= η7i
1 - ðπ 1i þ π 2i þ π 3i þ π 4i þ π 5i þ π 6i þ π 7i Þ
π 1i þ π 2i þ π 3i þ π 4i þ π 5i þ π 6i þ π 7i þ π 8i
= η8i
1 - ðπ 1i þ π 2i þ π 3i þ π 4i þ π 5i þ π 6i þ π 7i þ π 8i Þ
The following GLIMMIX commands fit a cumulative logit model with an ordinal
multinomial response.
proc glimmix ;
class id additive scale;
model scale(order=data)= additive/dist=Multinomial link=clogit
solution oddsratio;
estimate 'c=1, a=1' intercept 1 0 0 0 0 0 0 0 additive 1 0 0 0,
'c=2, a=1' intercept 0 1 0 0 0 0 0 0 additive 1 0 0 0,
'c=3, a=1' intercept 0 0 1 0 0 0 0 0 additive 1 0 0 0,
'c=4, a=1' intercept 0 0 0 1 0 0 0 0 additive 1 0 0 0,
8.6
Generalized Logit Models: Nominal Response Variables
Table 8.25 Fixed effects
tests in the multinomial
cumulative logit model
Type III tests of fixed effects
Den DF
Effect
Num DF
Additive
3
197
355
F-value
38.11
Pr > F
<0.0001
'c=5, a=1' intercept 0 0 0 0 1 0 0 0 additive 1 0 0 0,
'c=6, a=1' intercept 0 0 0 0 0 1 0 0 additive 1 0 0 0,
'c=7, a=1' intercept 0 0 0 0 0 0 1 0 additive 1 0 0 0,
'c=8, a=1' intercept 0 0 0 0 0 0 0 1 additive 1 0 0 0,
'c=1, a=2' intercept 1 0 0 0 0 0 0 0 additive 0 1 0 0,
'c=2, a=2' intercept 0 1 0 0 0 0 0 0 additive 0 1 0 0,
'c=3, a=2' intercept 0 0 1 0 0 0 0 0 additive 0 1 0 0,
'c=4, a=2' intercept 0 0 0 1 0 0 0 0 additive 0 1 0 0,
'c=5, a=2' intercept 0 0 0 0 1 0 0 0 additive 0 1 0 0,
'c=6, a=2' intercept 0 0 0 0 0 1 0 0 additive 0 1 0 0,
'c=7, a=2' intercept 0 0 0 0 0 0 1 0 additive 0 1 0 0,
'c=8, a=2' intercept 0 0 0 0 0 0 0 1 additive 0 1 0 0,
'c=1, a=3' intercept 1 0 0 0 0 0 0 0 additive 0 0 1 0,
'c=2, a=3' intercept 0 1 0 0 0 0 0 0 additive 0 0 1 0,
'c=3, a=3' intercept 0 0 1 0 0 0 0 0 additive 0 0 1 0,
'c=4, a=3' intercept 0 0 0 1 0 0 0 0 additive 0 0 1 0,
'c=5, a=3' intercept 0 0 0 0 1 0 0 0 additive 0 0 1 0,
'c=6, a=3' intercept 0 0 0 0 0 1 0 0 additive 0 0 1 0,
'c=7, a=3' intercept 0 0 0 0 0 0 1 0 additive 0 0 1 0,
'c=8, a=3' intercept 0 0 0 0 0 0 0 1 additive 0 0 1 0,
'c=1, a=4' intercept 1 0 0 0 0 0 0 0 additive 0 0 0 1,
'c=2, a=4' intercept 0 1 0 0 0 0 0 0 additive 0 0 0 1,
'c=3, a=4' intercept 0 0 1 0 0 0 0 0 additive 0 0 0 1,
'c=4, a=4' intercept 0 0 0 1 0 0 0 0 additive 0 0 0 1,
'c=5, a=4' intercept 0 0 0 0 1 0 0 0 additive 0 0 0 1,
'c=6, a=4' intercept 0 0 0 0 0 1 0 0 additive 0 0 0 1,
'c=7, a=4' intercept 0 0 0 0 0 0 1 0 additive 0 0 0 1,
'c=8, a=4' intercept 0 0 0 0 0 0 0 1 additive 0 0 0 1/ilink;
freq freq;
run;
Part of the results is shown in Table 8.25. The results of the analysis of variance
show that the type of additive used in the manufacture of cheese significantly affects
the degree of consumer acceptance (P = 0.0001). That is, the type of additive affects
the sensory characteristics of the cheese.
The contrast of hypothesis are presented in Table 8.26. The hypothesis tests are as
follows:
π additivei = π additivej ; 8i ≠ j
The results show that the additives provide different sensory characteristics that
are reflected in the evaluation of preference.
With the “solution” option in the model specification, Table 8.27 (fixed parameter
estimates) shows the solution of the maximum likelihood estimates for the fixed
356
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.26 Contrast of hypothesis in the acceptance of cheese made with four additives
Contrasts
Label
Additive effect 1 vs. 2
Additive effect 1 vs. 3
Additive effect 2 vs. 3
Additive effect 2 vs. 4
Additive effect 3 vs. 4
Num DF
1
1
1
1
1
Den DF
197
197
197
197
197
F-value
61.13
21.19
19.14
108.45
62.04
Pr > F
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Table 8.27 Maximum likelihood estimates of the fixed effects in the preference ratings of cheese
made with different types of additives in the multinomial cumulative logit model
Fixed parameter estimates
Effect
escala Additive
Intercept η1
1
Intercept η2
2
3
Intercept η3
4
Intercept η4
5
Intercept η5
Intercept η6
6
7
Intercept η7
8
Intercept η8
1
Aditivo α1
Aditivo α2
2
3
Aditivo α3
4
Aditivo α4
Estimate
-7.0802
-6.0250
-4.9254
-3.8568
-2.5206
-1.5685
-0.06688
1.4930
1.6128
4.9646
3.3227
0
Standard error
0.5640
0.4764
0.4257
0.3880
0.3453
0.3122
0.2738
0.3357
0.3805
0.4767
0.4218
.
DF
197
197
197
197
197
197
197
197
197
197
197
.
t-value
-12.55
-12.65
-11.57
-9.94
-7.30
-5.02
-0.24
4.45
4.24
10.41
7.88
.
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
0.8073
<0.0001
<0.0001
<0.0001
<0.0001
.
effects parameters. In this table, we observe the values of the estimated intercepts:
η1 = - 7:0802 defines the boundary between categories “1” and “2,” whereas
η2 = - 6:0250 defines the boundary between categories “2” and “3.” The third
intercept, ^η3 = - 4:9254, defines the boundary between categories “3” and “4”
and so forth. The estimated effects of the additive type ðαi , i = 1, 2, 3, and 4Þ are
1.628, 4.9646, 3.3227, and 0, respectively. From these values, linear predictors are
estimated for each of the categories.
For example, the estimated linear predictor for a cheese made with additive 1, where
the evaluator (consumer) assigns it category “1 = highly undesirable,” is represented as
η11 = η1 þ α1 = - 7:0802 þ 1:6128 = - 5:4674 ; for the category “2 = moderately
undesirable,” it is η21 = η2 þ α1 = - 6:0250 þ 1:6128 = - 4:4122; for the category
“3 = slightly undesirable,” it is η31 = η3 þ α1 = - 4:9254 þ 1:6128 = - 3:3126; and
for the category “4 = neither undesirable nor desirable,” it is
η41 = η4 þ α1 = - 3:8568 þ 1:6128 = - 2:2440. These values are shown in the “Estimate” column of Table 8.28; other categories are similarly calculated for each type of
additive.
The estimated values in Table 8.27 obtained with the “estimate” command in
conjunction with the “ilink” option prompts GLIMMIX to calculate the values of the
8.7
Exercises
357
linear predictors ηCi tabulated in the “Estimate” column and estimated probabilities
π Ciof all categories of each treatment, tabulated in the “Mean” column π cij , except
for the reference category.
From Table 8.28 (Estimates), we obtain the probabilities for each category that is
reported under the “Mean” column. In this case, the probability for π 11 = 0:004205.
This value is obtained by taking the inverse value of the linear predictor η11 = - 5:4674
π 11 = 1= 1 þ exp ð5:4674Þ = 0:004205 . To calculate the probability that a panelist
would assign a rating of 2 (2 = moderately undesirable) to cheese made with additive
1, we use the cumulative probability of 0.01198, which corresponds to π^21 þ π^11 . From
this value, we obtain π 21 = 0:01198 - π 11 = 0:01198 - 0:004205 = 0:007775 and for
the probability of assigning a rating of 3 to cheese made with additive
1, π 31 = 0:03514 - ðπ 21 þ π 11 Þ = 0:03514 - 0:001198 = 0:033942: Following the
same procedure, we obtain the other probabilities for the rest of the categories of each
of the additives used in the manufacturing of cheese, which are tabulated in Table 8.29
and can be seen in Fig. 8.3.
Figure 8.3 shows the probability results of each flavor rating for each of the
additives (it should be noted that some probability values were suppressed to avoid
overwriting). It can be seen that additive 1 primarily receives ratings of 5–7; additive
2 primarily receives ratings of 2–5; additive 3 primarily receives ratings of 4–6; and
additive 4 primarily receives ratings of 7–9.
The odds ratio results (Table 8.30) show the preferences more clearly. For
example, the odds ratio additive 1 vs. 4 states that the first additive is 5.017 times
more likely to receive a lower score than the fourth additive.
8.7
Exercises
Exercise 8.7.1 The dataset for this exercise corresponds to the results of 9 judges
who rated 2 classes of wine, namely, white wine (WW = 1) and red wine (RW = 2),
and, within each wine class, they rated 10 wines on a scale of 1–20 points. The
minimum rating for a particular wine was 7, and the maximum rating was 19.5. For
didactic purposes, ratings between 7 and 11 were assigned low quality, a rating
between 12 and 15 as medium quality, and anything above 15 was considered
excellent quality. The data are shown in Table 8.31 of the wine evaluation experiment under columns “Judge” (wine evaluator panelist), “Wine_type” (white wine:
1, red wine: 2), “Quality” (low, medium, and excellent), and the frequency of the
observed qualities (“y”).
(a) Fit the cumulative logit proportional odds model to these data. Perform a
complete and appropriate analysis of the data, focusing on:
(i) An evaluation of the effects of the combination of treatments
(ii) Interpretation of the odds ratios
(iii) The expected probability per category for each treatment
358
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.28 Estimates on the model scale (Estimate) and on the data scale (Mean) based on judges’
preference ratings of cheese made with different types of additives in the multinomial cumulative
logit model
Estimates
Label
c = 1,
a=1
c = 2,
a=1
c = 3,
a=1
c = 4,
a=1
c = 5,
a=1
c = 6,
a=1
c = 7,
a=1
c = 8,
a=1
c = 1,
a=2
c = 2,
a=2
c = 3,
a=2
c = 4,
a=2
c = 5,
a=2
c = 6,
a=2
c = 7,
a=2
c = 8,
a=2
c = 1,
a=3
c = 2,
a=3
c = 3,
a=3
c = 4,
a=3
c = 5,
a=3
Estimate
-5.4674
Standard
error
0.5236
DF
197
t-value
-10.44
Pr > |t|
<0.0001
Mean
0.004205
Standard error
mean
0.002192
-4.4122
0.4278
197
-10.31
<0.0001
0.01198
0.005064
-3.3126
0.3700
197
-8.95
<0.0001
0.03514
0.01255
-2.2440
0.3267
197
-6.87
<0.0001
0.09587
0.02832
-0.9078
0.2833
197
-3.20
0.0016
0.2875
0.05804
0.04425
0.2646
197
0.17
0.8673
0.5111
0.06611
1.5459
0.3017
197
5.12
<0.0001
0.8243
0.04369
3.1058
0.4057
197
7.65
<0.0001
0.9571
0.01665
-2.1155
0.4106
197
-5.15
<0.0001
0.1076
0.03942
-1.0603
0.3009
197
-3.52
0.0005
0.2572
0.05749
0.03922
0.2735
197
0.14
0.8861
0.5098
0.06836
1.1078
0.2969
197
3.73
0.0002
0.7517
0.05542
2.4441
0.3397
197
7.19
<0.0001
0.9201
0.02497
3.3961
0.3724
197
9.12
<0.0001
0.9676
0.01168
4.8978
0.4249
197
11.53
<0.0001
0.9926
0.003124
6.4576
0.5045
197
12.80
<0.0001
0.9984
0.000789
-3.7575
0.4761
197
-7.89
<0.0001
0.02281
0.01061
-2.7023
0.3677
197
-7.35
<0.0001
0.06284
0.02165
-1.6027
0.3001
197
-5.34
<0.0001
0.1676
0.04186
-0.5341
0.2556
197
-2.09
0.0379
0.3696
0.05955
0.8021
0.2610
197
3.07
0.0024
0.6904
0.05579
(continued)
8.7
Exercises
359
Table 8.28 (continued)
Estimates
Label
c = 6,
a=3
c = 7,
a=3
c = 8,
a=3
c = 1,
a=4
c = 2,
a=4
c = 3,
a=4
c = 4,
a=4
c = 5,
a=4
c = 6,
a=4
c = 7,
a=4
c = 8,
a=4
Standard
error
0.2984
DF
197
t-value
5.88
Pr > |t|
<0.0001
Mean
0.8525
Standard error
mean
0.03752
3.2558
0.3618
197
9.00
<0.0001
0.9629
0.01293
4.8157
0.4528
197
10.63
<0.0001
0.9920
0.003610
-7.0802
0.5640
197
-12.55
<0.0001
0.000841
0.000474
-6.0250
0.4764
197
-12.65
<0.0001
0.002412
0.001146
-4.9254
0.4257
197
-11.57
<0.0001
0.007207
0.003046
-3.8568
0.3880
197
-9.94
<0.0001
0.02070
0.007865
-2.5206
0.3453
197
-7.30
<0.0001
0.07443
0.02379
-1.5685
0.3122
197
-5.02
<0.0001
0.1724
0.04455
-0.06688
0.2738
197
-0.24
0.8073
0.4833
0.06838
1.4930
0.3357
197
4.45
<0.0001
0.8165
0.05029
Estimate
1.7541
Exercise 8.7.2 Data were obtained from a series of experiments conducted to
reduce damage to potato tubers due to a potato lifter. The experiments were
conducted at the Institute of Agricultural Engineering (IMAG-DLO) in Wageningen,
the Netherlands. One source of damage was the type of rod used in the lifter. In the
experiment – under consideration – eight types of rods were compared. It is an
empirical fact that the degree of damage varies considerably between potato varieties
with the type of rope used in the lifting of full potato sacks. Three blocks of
observations were obtained for the combinations of varieties and rope types. Most
of the combinations involved about 20 potatoes. For some combinations, there are
no data due to an insufficient number of large potatoes. Tubers were dropped from a
given height. To determine the damage, all tubers were peeled and the degree of blue
coloration was classified into one of four classes (class 1: no damage; class 2: slight
damage; class 3: moderate damage; and class 4: severe damage). The observations,
in the form of counts per class and combination, are shown in Table 8.32 of the tuber
experiment whose columns are “Variety” (1, 2, 3, 4, 4, 5, 6), “String” (1, 2, 3, 4, 5, 6,
7, 8), “Block” (1, 2, 3), Type of damage (sd = no damage, dl = light damage,
dm = moderate damage, ds = severe damage), and the observed frequency (Y ).
Cal2
0.00778
0.1496
0.04003
0.00157
Cal3
0.02316
0.2526
0.10476
0.0048
Note: Grade1 = Grade 1, Grade2 = Grade 2,...., Grade9 = Grade 9
Additive 1
Additive 2
Additive 3
Additive 4
Rating
Cal1
0.00421
0.1076
0.02281
0.00084
Cal4
0.06073
0.2419
0.202
0.01349
Cal5
0.19163
0.1684
0.3208
0.05373
Cal6
0.2236
0.0475
0.1621
0.09797
Cal7
0.3132
0.025
0.1104
0.3109
Cal8
0.1328
0.0058
0.0291
0.3332
Cal9
0.0429
0.0016
0.008
0.1835
8
Table 8.29 Probabilities calculated for each of the ratings by additives used in the manufacture of cheese
360
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Exercises
Probability of acceptability
8.7
361
1.0
0.0429
0.
0.9
0.1328
0.
0.
0.0291
0.1104
0.
0.0475
0.
0.1835
0.
0.
0.1684
0.8
0.1621
0.
0.7
0.3132
0.
0.3332
0.
0.
0.2419
0.6
0.
0.3208
0.5
0.4
0.2236
0.
0.
0.2526
0.3
0.3109
0.
0. 02
0.202
0.2
0.
0.19163
63
0.1
0.
0.1076
Aditivo 1
Aditivo 2
0.0
Cal1
Cal2
0.
0.09797
97
0.05373
0.
73
0.
0.10476
76
0.
0.06073
73
Cal3
Cal4
Cal5
0.
0.02281
81
Aditivo 3
Cal6
Cal7
Aditivo 4
Cal8
Cal9
Fig. 8.3 Estimated probabilities for the categories of acceptability for the cheese according to the
type of additive
Table 8.30 Odds ratio
Odds ratio estimates
Aditivo
_Aditivo
1
4
2
4
3
4
Estimate
5.017
143.257
27.735
DF
197
197
197
95% Confidence limits
2.369
10.625
55.953
366.783
12.071
63.724
(a) List the components of the multinomial GLMM.
(b) Fit the cumulative logit proportional odds model to these data. Perform a
complete and appropriate analysis of the data, focusing on:
(i) An evaluation of the effects of the combination of treatments
(ii) Interpretation of the odds ratios
(iii) The expected probability per category for each treatment
(c) Test whether the proportional odds assumption is viable. Cite relevant evidence
to support your conclusion regarding the adequacy of the assumption.
(d) If as a result of b), you consider that an alternative cumulative logit model is
better, then revise your analysis in a) accordingly.
Exercise 8.7.3 Fit a generalized multinomial logit model using the dataset from
Exercise 8.7.2 of this section, following the instructions:
362
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.31 Results of the wine evaluation experiment
Judge
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
Wine_type
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
Quality
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
y
4
6
0
3
6
1
3
5
2
2
7
1
0
4
6
0
1
9
1
3
6
4
3
3
6
3
1
3
5
2
0
4
6
0
6
4
1
9
0
3
5
2
(continued)
8.7
Exercises
363
Table 8.31 (continued)
Judge
8
8
8
8
8
8
9
9
9
9
9
9
Wine_type
1
1
1
2
2
2
1
1
1
2
2
2
Quality
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
Low
Medium
Excellent
y
0
7
3
0
6
4
0
5
5
2
4
4
(a) List the components of this model.
(b) Perform a thorough and appropriate analysis of the data, focusing on:
(i) An evaluation of the main effects and treatment interaction
(ii) Odds ratio interpretation
(iii) The expected probability per category for each treatment
(c) Comment on and discuss your results. Cite relevant evidence to support your
conclusion regarding the adequacy of the assumption.
Exercise 8.7.4 In this exercise, the effects of judges’ experience on quality ratings
of canned beans are assessed. Canning quality is one of the most essential traits
required in all new dry bean (Phaseolus vulgaris L.) varieties, and selection for this
trait is a critical part of bean breeding programs. Advanced lines, which are candidates for release as varieties, must be evaluated for canning quality for at least
3 years from samples grown at different locations. Quality is evaluated by a panel of
judges with varying levels of experience in evaluating breeding lines for visual
quality traits. In all, 264 bean breeding lines from 4 commercial classes were
conserved according to the procedures described by Walters et al. (1997).
These included 62 white (navy), 65 black, 55 kidney, and 82 pinto bean lines plus
control lines and “checks.” The visual appearance of the processed beans was
determined subjectively by a panel of 13 judges on a 7-point hedonic scale
(1 = very undesirable, 4 = neither desirable nor undesirable, 7 = very desirable).
The beans were presented to the panel of judges in random order at the same time.
Prior to evaluating the samples, all judges were shown examples of samples rated as
satisfactory (4). There is concern that certain judges, due to lack of experience, may
not be able to score canned samples correctly.
From attribute-based product evaluations, inferences about the effects of experience can be drawn from the psychology literature (Wallsten and Budescu (1981).
Prior to the bean canning quality rating experiment, it was postulated that not only do
364
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.32 Results of the tuber experiment. V = variety, C = string, B = block, D = damage
(sd = no damage, dl = slight damage, dm = moderate damage, ds = severe damage), and
Y = observed frequency
V
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
C
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
2
3
B
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
D
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
Y
5
14
1
0
6
11
1
0
6
13
0
0
2
9
6
3
11
8
0
0
12
5
2
0
8
12
0
0
12
4
0
0
5
31
2
1
6
11
1
0
5
V
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
C
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
2
3
B
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
D
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
Y
4
5
4
7
8
3
0
0
3
10
6
1
1
3
11
5
16
3
1
0
16
3
0
0
11
9
0
0
10
10
0
0
5
7
5
1
13
6
1
0
5
V
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
C
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
2
3
B
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
D
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
Y
3
2
8
7
18
1
0
0
5
7
4
4
1
4
6
9
12
7
1
0
16
3
0
1
20
0
0
0
18
2
0
0
6
8
5
1
12
6
1
1
10
(continued)
8.7
Exercises
365
Table 8.32 (continued)
V
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
C
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
B
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
D
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
Y
13
0
0
2
11
9
0
16
4
0
0
15
5
0
0
9
7
3
0
0
0
0
0
0
0
0
1
7
10
2
0
1
19
0
0
4
13
3
0
15
4
0
V
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
C
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
B
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
D
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
Y
12
3
0
0
8
11
1
10
9
1
0
10
7
1
1
11
6
1
0
13
6
0
0
9
7
2
0
18
2
0
0
13
6
0
0
0
9
9
2
16
2
1
V
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
C
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
B
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
D
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
Y
8
0
2
2
6
10
2
16
4
0
0
14
5
0
0
16
3
0
0
16
3
0
1
3
15
2
0
16
6
2
2
8
9
2
1
3
5
10
1
16
3
0
(continued)
366
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.32 (continued)
V
1
1
1
1
1
1
1
1
1
1
1
1
1
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
C
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
B
3
3
3
3
3
3
3
3
3
3
3
3
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
D
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
Y
1
11
4
0
5
11
9
0
0
17
2
1
0
4
9
7
0
12
8
0
0
10
10
0
0
2
8
4
6
17
2
0
0
18
2
0
0
19
1
0
0
15
V
2
2
2
2
2
2
2
2
2
2
2
2
2
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
C
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
B
3
3
3
3
3
3
3
3
3
3
3
3
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
D
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
Y
0
15
3
2
0
14
5
1
0
12
3
2
1
9
10
1
0
17
2
0
0
15
5
0
0
12
7
0
0
16
4
0
0
20
0
0
0
20
0
0
0
20
V
3
3
3
3
3
3
3
3
3
3
3
3
3
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
C
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
B
3
3
3
3
3
3
3
3
3
3
3
3
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
D
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
Y
1
15
5
0
0
15
5
0
0
16
4
0
0
5
14
1
0
18
2
0
0
5
14
0
0
5
14
1
0
15
5
0
0
19
1
0
0
18
2
0
0
18
(continued)
8.7
Exercises
367
Table 8.32 (continued)
V
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
C
8
8
8
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
B
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
D
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
Y
3
0
1
4
11
3
1
17
2
0
1
10
9
0
0
4
8
6
1
18
2
0
0
19
1
0
0
20
0
0
0
20
0
0
0
10
9
1
0
16
3
1
V
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
C
8
8
8
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
B
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
D
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
Y
0
0
0
10
10
0
0
19
1
0
0
14
6
0
0
7
11
1
0
15
4
1
0
19
1
0
0
17
2
0
0
18
2
0
0
5
11
4
0
12
8
0
V
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
C
8
8
8
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
1
1
1
1
2
2
2
B
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
D
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
Y
2
0
0
6
13
0
0
18
2
0
0
13
7
0
0
0
15
5
0
16
3
0
0
19
1
0
0
17
3
0
0
15
4
0
0
3
15
2
0
13
7
0
(continued)
368
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.32 (continued)
V
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
C
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
B
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
D
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
Y
0
15
4
0
0
5
11
4
0
17
2
0
0
16
2
0
0
17
2
1
0
17
2
0
0
V
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
C
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
B
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
D
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
Y
0
7
12
1
0
6
6
6
2
16
4
0
0
17
3
0
0
18
2
0
0
17
2
0
0
V
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
C
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
B
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
D
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
sd
dl
dm
ds
Y
0
2
18
0
0
4
10
6
0
6
13
1
0
18
2
0
0
15
5
0
0
19
1
0
0
less experienced judges have a more severe rating than do more experienced judges
but also that experience should have little or no effect on the white beans for which
the canning procedure was developed. Judges are stratified for the purpose of
analysis by experience (less than 5 years, greater than 5 years).
Counts by canning quality, judge experience, and bean breeding lines are listed in
the following table (Table 8.33).
8.7
Exercises
369
Table 8.33 Bean experiment results
Cal
1
2
3
4
5
6
7
Black
<
5 Years
13
91
123
72
24
2
0
>
5 Years
32
78
124
122
31
3
0
Kidney
<
5 Years
7
32
136
101
47
6
1
>
5 Years
10
31
96
104
71
18
0
Navies
<
5 Years
10
56
84
84
51
24
1
>
5 Years
22
51
107
98
52
37
5
Pinto
<
5 Years
13
29
91
109
60
25
1
>
5 Years
2
17
68
124
109
78
12
(a) Fit the generalized logit model to these data. Perform a complete and appropriate
analysis of the data, focusing on:
(i) An evaluation of the effects of the combination of treatments
(ii) Interpretation of the odds ratios
(iii) The expected probability per category for each treatment
(b) Test whether the proportional odds assumption is viable. Cite relevant evidence
to support your conclusion regarding the adequacy of the assumption.
Exercise 8.7.5 An experiment was conducted to look at the damage levels (ordinal
categories 0–4) of Picea sitchensis shoots in two time periods (10 November and
8 December), at four temperatures (different on each date), and at four ozone levels
(Table 8.34).
(a) Fit the cumulative logit proportional odds model to these data. Perform a
complete and appropriate analysis of the data, focusing on:
(i) An evaluation of the effects of the combination of treatments
(ii) Interpretation of the odds ratios
(iii) The expected probability per category for each treatment
(b) Test whether the proportional odds assumption is viable. Cite relevant evidence
to support your conclusion regarding the adequacy of the assumption.
370
8 Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.34 Experimental results of Picea sitchensis sprouts
Weather
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Temperature (°C)
-9
-9
-9
-9
-9
-12
-12
-12
-12
-12
-15
-15
-15
-15
-15
-18
-18
-18
-18
-18
-9
-9
-9
-9
-9
-12
-12
-12
-12
-12
-15
-15
-15
-15
-15
-18
-18
-18
-18
-18
-9
-9
Ozone
170
170
170
170
170
170
170
170
170
170
170
170
170
170
170
170
170
170
170
170
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
70
70
Category
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
Frequency
1
10
2
2
0
0
8
3
1
3
0
3
2
4
6
0
1
1
4
9
1
9
4
1
0
0
7
7
1
0
0
1
5
6
3
0
0
4
5
6
4
6
(continued)
8.7 Exercises
371
Table 8.34 (continued)
Weather
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
Temperature (°C)
-9
-9
-9
-12
-12
-12
-12
-12
-15
-15
-15
-15
-15
-18
-18
-18
-18
-18
-9
-9
-9
-9
-9
-12
-12
-12
-12
-12
-15
-15
-15
-15
-15
-18
-18
-18
-18
-18
-15
-15
-15
-15
Ozone
70
70
70
70
70
70
70
70
70
70
70
70
70
70
70
70
70
70
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
170
170
170
170
Category
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
Frequency
3
2
0
1
6
6
2
0
0
3
6
4
2
0
1
0
5
9
2
11
2
0
0
1
6
6
2
0
2
4
4
3
2
0
4
3
5
3
3
8
4
1
(continued)
372
8 Generalized Linear Mixed Models for Categorical and Ordinal Responses
Table 8.34 (continued)
Weather
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Temperature (°C)
-15
-19
-19
-19
-19
-19
-23
-23
-23
-23
-23
-27
-27
-27
-27
-27
-15
-15
-15
-15
-15
-19
-19
-19
-19
-19
-23
-23
-23
-23
-23
-27
-27
-27
-27
-27
-15
-15
-15
-15
-15
-19
Ozone
170
170
170
170
170
170
170
170
170
170
170
170
170
170
170
170
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
120
70
70
70
70
70
70
Category
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
Frequency
3
0
10
5
0
4
0
1
8
4
6
0
0
2
3
14
6
6
8
0
0
1
12
7
0
0
0
0
7
7
6
0
0
1
2
17
9
4
5
2
0
2
(continued)
8.7
Exercises
373
Table 8.34 (continued)
Weather
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Temperature (°C)
-19
-19
-19
-19
-23
-23
-23
-23
-23
-27
-27
-27
-27
-27
-15
-15
-15
-15
-15
-19
-19
-19
-19
-19
-23
-23
-23
-23
-23
-27
-27
-27
-27
-27
Ozone
70
70
70
70
70
70
70
70
70
70
70
70
70
70
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Category
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
Frequency
10
6
0
2
0
3
5
4
8
0
0
0
3
17
5
6
3
1
2
6
5
3
1
2
0
4
2
1
3
0
1
0
5
11
374
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Appendix
Data: CRD with a multinomial response: ordinal
Rep
Trt
Cat
Freq
rep1
M1H1
Without
0
rep2
M1H1
Without
2
rep3
M1H1
Without
2
rep4
M1H1
Without
2
rep1
M1H2
Without
2
rep2
M1H2
Without
0
rep3
M1H2
Without
4
rep4
M1H2
Without
2
rep1
M1H3
Without
3
rep2
M1H3
Without
7
rep3
M1H3
Without
1
rep4
M1H3
Without
2
rep1
M1H4
Without
0
rep2
M1H4
Without
5
rep3
M1H4
Without
2
1
rep4
M1H4
Without
rep1
M1H1
Moderate
3
rep2
M1H1
Moderate
2
rep3
M1H1
Moderate
3
rep4
M1H1
Moderate
5
rep1
M1H2
Moderate
3
rep2
M1H2
Moderate
3
rep3
M1H2
Moderate
6
rep4
M1H2
Moderate
3
rep1
M1H3
Moderate
4
rep2
M1H3
Moderate
2
rep3
M1H3
Moderate
1
rep4
M1H3
Moderate
3
rep1
M1H4
Moderate
5
rep2
M1H4
Moderate
4
rep3
M1H4
Moderate
8
rep4
M1H4
Moderate
4
M1H1
Severe
6
rep1
rep2
M1H1
Severe
6
rep3
M1H1
Severe
5
rep4
M1H1
Severe
3
rep1
M1H2
Severe
5
rep2
M1H2
Severe
7
rep3
M1H2
Severe
0
rep4
M1H2
Severe
5
Rep
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
Trt
M2H3
M2H3
M2H3
M2H3
M2H4
M2H4
M2H4
M2H4
M2H1
M2H1
M2H1
M2H1
M2H2
M2H2
M2H2
M2H2
M2H3
M2H3
M2H3
M2H3
M2H4
M2H4
M2H4
M2H4
M3H1
M3H1
M3H1
M3H1
M3H2
M3H2
M3H2
M3H2
M3H3
M3H3
M3H3
M3H3
M3H4
M3H4
M3H4
M3H4
Cat
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Without
Without
Without
Without
Without
Without
Without
Without
Without
Without
Without
Without
Without
Without
Without
Without
Freq
3
1
3
2
4
2
2
5
4
6
7
4
5
2
3
4
3
4
4
4
5
6
0
3
0
3
2
0
5
3
3
2
0
2
1
0
3
5
7
3
(continued)
Appendix
Data: CRD with a multinomial response: ordinal
Rep
Trt
Cat
Freq
rep1
M1H3
Severe
3
rep2
M1H3
Severe
1
rep3
M1H3
Severe
7
rep4
M1H3
Severe
5
rep1
M1H4
Severe
5
rep2
M1H4
Severe
1
rep3
M1H4
Severe
0
rep4
M1H4
Severe
5
rep1
M2H1
Without
1
rep2
M2H1
Without
2
rep3
M2H1
Without
1
rep4
M2H1
Without
1
rep1
M2H2
Without
1
rep2
M2H2
Without
3
rep3
M2H2
Without
1
rep4
M2H2
Without
4
rep1
M2H3
Without
4
rep2
M2H3
Without
5
rep3
M2H3
Without
3
rep4
M2H3
Without
4
rep1
M2H4
Without
1
rep2
M2H4
Without
1
rep3
M2H4
Without
8
rep4
M2H4
Without
2
rep1
M2H1
Moderate
4
rep2
M2H1
Moderate
2
rep3
M2H1
Moderate
2
rep4
M2H1
Moderate
5
rep1
M2H2
Moderate
4
rep2
M2H2
Moderate
4
rep3
M2H2
Moderate
6
rep4
M2H2
Moderate
2
375
Rep
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
rep1
rep2
rep3
rep4
Trt
M3H1
M3H1
M3H1
M3H1
M3H2
M3H2
M3H2
M3H2
M3H3
M3H3
M3H3
M3H3
M3H4
M3H4
M3H4
M3H4
M3H1
M3H1
M3H1
M3H1
M3H2
M3H2
M3H2
M3H2
M3H3
M3H3
M3H3
M3H3
M3H4
M3H4
M3H4
M3H4
Cat
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Severe
Freq
0
5
5
0
3
2
6
1
3
5
3
3
0
2
3
4
9
2
3
10
2
5
1
7
6
3
6
7
7
3
0
3
376
8
Generalized Linear Mixed Models for Categorical and Ordinal Responses
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0
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the copyright holder.
Chapter 9
Generalized Linear Mixed Models
for Repeated Measurements
9.1
Introduction
Repeated measures data, also known as longitudinal data, are those derived from
experiments in which observations are made on the same experimental units at
various planned times. These experiments can be of the regression or analysis of
variance (ANOVA) type, can contain two or more treatments, and are set up using
familiar designs, such as completely randomized design (CRD), randomized complete block design (RCBD), or randomized incomplete blocks, if blocking is appropriate, or using row and column designs such as Latin squares when appropriate.
Repeated measures designs are widely used in the biological sciences and are fairly
well understood for normally distributed data but less so with binary, ordinal, count
data, and so on. Nevertheless, recent developments in statistical computing methodology and software have greatly increased the number of tools available for
analyzing categorical data.
A generalized linear mixed model (GLMM) is one of the most useful and
sophisticated structures in modern statistics, as it allows complex structures to be
incorporated into the framework of a general linear model. Fitting such models has
been the subject of much research over the last three decades. GLMMs, for repeated
measures, combine both generalized linear model (GLM) theory (e.g., a binomial,
multinomial, or Poisson response variable) and linear mixed effects models.
Experimentation is sometimes not well understood since researchers believe that
it involves only the manipulation of the levels of independent variables and the
observation of subsequent responses in dependent variables. Independent variables,
whose levels are determined or set by the experimenter, are said to have fixed effects,
although random effects are also very common, where the levels of the effects are
assumed to be randomly selected from an infinite population of possible levels.
Many variables of interest in research are not fully amenable to experimental
manipulation but can nevertheless be studied by considering them to have random
effects. For example, the genetic composition of individuals of a species cannot be
© The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8_9
377
378
9 Generalized Linear Mixed Models for Repeated Measurements
manipulated experimentally, but it is of great interest to geneticists aiming to assess
the genetic contribution to individual variation of some specific behaviors.
A GLMM with repeated measures is a generalization of the standard linear
model, and this generalization is due to (1) the presence of more than one response
variable that can be binary, ordinal, count, and so on and (2) the nonconstant
correlation and/or variability exhibited by the data. The linear mixed model, therefore, gives you the flexibility to model not only the means of your data (as in the
standard linear model) but also their variances and covariances. Usually, a normal
distribution is assumed for random effects. Since normally distributed data can be
modeled entirely in terms of their means and variances/covariances, the two sets of
parameters in a linear mixed model actually specify the full probability distribution
of the data. The parameters of the mean structure in the model are called (known as)
fixed effects parameters, which can be qualitative (as in traditional analysis of
variance) or quantitative (as in standard regression), and the parameters of the
variance–covariance of the model are known as covariance parameters, which help
distinguish a linear mixed model from the standard linear model. Covariance
parameters come up quite frequently in the following applications, with two more
typical scenarios:
(a) Experimental units on which data are measured can be grouped into clusters, and
data from a common cluster are correlated.
(b) Repeated measurements of the same experimental unit are taken, and these
repeated measurements correlate or show some variability.
The first scenario can be generalized to include a set of clusters nested within one
another. For example, if students are the experimental unit, they can be grouped into
classes (clusters), which, in turn, can be grouped into schools. Each level of this
hierarchy may present an additional source of variability and correlation. The second
scenario occurs in longitudinal studies, in which repeated measurements of the same
experimental unit over time are taken. Alternatively, these repeated measures could
be spatial or multivariate.
9.2
Example of Turf Quality
The proportional odds model, introduced by McCullagh (1980), was proposed as an
extension of the generalized linear model used for ordinal responses. One can recall
that the proportional odds model is a special case of a GLM with a cumulative link
function in which the probability of an observation falling into a category or below is
modeled. In the case of a logit link, with only two categories (a binary response), the
proportional odds model reduces to a standard logistic regression or a classification
model. As with any other type of response variable, repeated measurements are
common in agronomic research. They result in clustered data structures with correlations between repeated observations in the same experimental unit that must be
taken into account in the analysis.
9.2
Example of Turf Quality
379
Table 9.1 Turf quality of five grass varieties (low, Med = medium, Excel = Excellent,
Sept = September)
Variety
1
2
3
4
5
No. of plots
18
17
17
18
18
May
Low
4
2
2
8
1
Med
10
11
11
7
11
Excel
4
4
4
3
6
July
Low
1
0
2
4
3
Med
9
7
8
8
4
Excel
8
10
7
6
11
Sept
Low
0
0
2
4
3
Med
12
9
11
13
6
Excel
6
8
4
1
9
The data were obtained from an experiment studying the turf quality of five grass
varieties. The varieties were sown independently in 17 or 18 plots. The evaluations
of the plots (experimental units) were carried out in the months of May, July, and
September of the growing season, and turf quality was classified on an ordinal scale
into three categories: low quality, medium quality, and excellent quality, as demonstrated in Table 9.1.
The components of the GLMM, with repeated measures with an ordinal multinomial response, are as follows:
Distributions: y1ij, y2ij, y3ij|ρij~Multinomial(Nij, π 1ij, π 2ij, π 3ij), where y1ij, y2ij, and y3ij
are the observed frequencies of the responses (turf quality) in each c category
(low, medium, and excellent), and ρij is the random effect due to the combination
variety × month (measurement time), assuming ρij N 0, σ 2ρ .
Linear predictor: η(c)ij = ηc + τi + ρij, where η(c)ij is the cth link (c = 1, 2) in the ijth
combination variety × month, ηc is the intercept for the cth link, τi is the fixed
effect due to the ith treatment, and ρij is the random effect due to the ijth
measurement of variety × month ρij N 0, σ 2variety × month
. The link functions
for each category are as follows:
log
log
π 0ij
= η0ij
1 - π 0ij
π 0ij þ π 1ij
1 - π 0ij þ π 1ij
= η1ij
The following Statistical Analysis Software (SAS) program fits a repeated measures GLMM with an ordinal response.
proc glimmix data=turfgrass method=laplace;
class Variety time;
model cat(order=data)=variety|time/dist=Multinomial link=clogit
solution oddsratio;
random intercept/subject=variety type=cs solution ;
380
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.2 Fit statistics under
different correlation structures
Fit statistics
-2 Log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (Consistent Akaike's
information criterion)
(smaller is better)
HQIC (Hannan Quinn
information criterion) (smaller is better)
Covariance structure
Toep
CS
AR(1) (1)
497.38 497.46 497.37
513.38 513.46 511.37
513.94 514.02 511.81
510.26 510.34 508.64
518.26 518.34 515.64
504.99
505.07
504.03
UN
n
o
c
o
n
v
e
r
g
e
estimate 'c=1, var=1' intercept 1 0 variedad 1 0 0 0 0,
'c=2, var=1' intercept 0 1 variedad 1 0 0 0 0,
'c=1, var=2' intercept 1 0 variedad 0 1 0 0 0,
'c=2, var=2' intercept 0 1 variedad 0 1 0 0 0,
'c=1, var=3' intercept 1 0 variedad 0 0 1 0 0,
'c=2, var=3' intercept 0 1 variedad 0 0 1 0 0,
'c=1, var=4' intercept 1 0 variedad 0 0 0 1 0,
'c=2, var=4' intercept 0 1 variedad 0 0 0 1 0,
'c=1, var=5' intercept 1 0 variedad 0 0 0 0 1,
'c=2, var=5' intercept 0 1 variedad 0 0 0 0 1/ilink;
freq y;
run;
Mixed models have advantages over fixed linear models (Littell et al. 1996)
because they have the ability to incorporate fixed (Xβ) and random effects (Zb)
that allow us to select different variance–covariance structures for repeated measures
experiments (with or without missing data) to see which covariance structure best fits
the model (Henderson 1984; Smith et al. 2005). Selecting or building a good enough
model involves selecting a covariance structure that best fits the dataset. The
information criteria minus two Restricted Log Likelihood (-2RLL), Akaike information criterion (AIC), Corrected Akaike’s information criterion (AICC), Bayesian
information criterion (BIC), etc.) provided by proc GLIMMIX are used as statistical
fit measures to select the variance structure (compound symmetry (“CS”), first-order
autoregressive (“AR(1)”), Toeplitz (“Toep(1)”), unstructured (“UN)”) that best
models the dataset.
Most of the commands have already been explained. To provide the correlation
structure that you want to model, with the above program, you vary the “TYPE”
option = (CS, AR(1), Toep(1), and UN) separately to specify each of the covariance
structures in the parentheses. Part of the results is shown below.
According to the fit statistics (Table 9.2), the covariance structure that best fits the
dataset is Toeplitz of order 1 (Toep(1)). The type III tests of fixed effects, shown in
Table 9.3 part (a), indicate that grass variety provides different turfgrass qualities
9.2
Example of Turf Quality
381
Table 9.3 Results of the analysis of variance
(a) Type III tests of fixed effects
Effect
Num degree of freedom (DF)
Variety
4
(b) Solutions for fixed effects
Effect
Cat
Variety
Estimate
Intercept
Low
-2.4509
Intercept
Medium
0.1961
Variety
Var1
0.4261
Variety
Var2
-0.01502
Variety
Var3
0.6125
Variety
Var4
1.4904
Variety
Var5
0
Den DF
10
Standard error
0.3219
0.2721
0.3753
0.3785
0.3825
0.3943
.
Pr > F
0.0202
F-value
4.80
DF
10
10
10
10
10
10
.
t-value
-7.61
0.72
1.14
-0.04
1.60
3.78
.
Pr > |t|
<0.0001
0.4875
0.2827
0.9691
0.1404
0.0036
.
Table 9.4 Estimated linear predictors and means on the model scale (Estimate) and on the data
scale (Mean) for observed turfgrass quality in grass varieties in the multinomial generalized logit
model
Estimates
Label
c = 1,
var = 1
c = 2,
var = 1
c = 1,
var = 2
c = 2,
var = 2
c = 1,
var = 3
c = 2,
var = 3
c = 1,
var = 4
c = 2,
var = 4
c = 1,
var = 5
c = 2,
var = 5
Standard
error
0.3018
DF
10
t-value
-6.71
Pr > |t|
<0.0001
Mean
0.1166
Standard error
mean
0.03110
0.6222
0.2646
10
2.35
0.0405
0.6507
0.06013
-2.4659
0.3177
10
-7.76
<0.0001
0.07828
0.02292
0.1811
0.2667
10
0.68
0.5125
0.5452
0.06613
-1.8384
0.3040
10
-6.05
0.0001
0.1372
0.03599
0.8086
0.2760
10
2.93
0.0150
0.6918
0.05884
-0.9605
0.2791
10
-3.44
0.0063
0.2768
0.05588
1.6865
0.2992
10
5.64
0.0002
0.8438
0.03944
-2.4509
0.3219
10
-7.61
<0.0001
0.07937
0.02352
0.1961
0.2721
10
0.72
0.4875
0.5489
0.06737
Estimate
-2.0248
(P = 0.0202). The “solution” option in the model specification “Model” provides the
solution of fixed effects of the model (intercepts and treatments), which we use to
^ i (part (b)).
estimate the linear predictors ^ηci = ^ηc þ Variety
The probabilities π ci obtained using the “Estimate” information are tabulated
under the “Mean” column of Table 9.4.
382
9
Generalized Linear Mixed Models for Repeated Measurements
From these values, we can observe that for the category "c = 1, var = 1, " the
value of the linear predictor is η11 = η1 þ variety1 = - 2:0248. Taking the inverse of
^η11 corresponds to the probability of π 11 = 0:1166 of observing “Low”-quality grass
of variety 1. Now, for the category "c = 2, var = 1, " the inverse of the linear
predictor is 0.6507, which is the estimate of the probability π 11 þ π 21 . From this
value, we can obtain the probability that variety 1 provides grass of “Medium”
quality, that is, π 11 þ π 21 = 0:6504, and, substituting the value of π 11 , we obtain the
probability value π 21 = 0:6507 - 0:1166 = 0:5341. With these two probability estimates π 11 and π 21 , it is possible to estimate the probability that variety 1 will yield an
“Excellent” quality turf, which is equal to π 31 = 1 - 0:6504 = 0:3496. Likewise, we
obtain the values of the remaining probabilities π ci for the rest of the grass varieties.
9.3
Effect of Insecticides on Aphid Growth
A cage experiment was used to investigate the effect of three insecticides on aphid
colonies with partial resistance to a common active compound. There were eight
treatments: all combinations of the three insecticides and a control (no insecticide)
with two types of colonies (susceptible or partially resistant). The experiment was
organized as an RCBD with six blocks of eight cages, and each cage was assigned a
treatment combination in each block. A colony of aphids was reared in each cage,
and the number of live aphids was recorded before insecticide treatment was applied
and then 2 and 6 days after application. Both hatches and deaths could occur within
each cage between evaluations. The dataset from this experiment is shown below
(Table 9.5).
Following the same reasoning as in previous examples, the components of the
GLMM with a Poisson response and repeated measures, which models the number
of aphids (yijkl), is described in the following lines.
Distributions: yijkl j bl , insecticide × cloneðblockÞijðlÞ Poisson λijkl
bl N 0, σ 2block , insecticide × cloneðblockÞijðlÞ N 0, σ 2insecticide × clone × block :
Linear predictor: ηijkl = θ + Ii + Cj + (IC)ij + bl + IC(b)ij(l ) + τl + (Iτ)il + (Cτ)il + (ICτ)ijkl,
Cj + (IC)ij + bl + IC(b)ij(l ) + τl + (Iτ)il + (Cτ)il + (ICτ)ijkl, where ηijkl is the linear
predictor, θ is the intercept, Ii (i = 1, 2, 3) is the fixed effect due to the insecticide,
Cj ( j = 1, 2) is the fixed effect due to the aphid clone, (IC)ij is the fixed effect due
to the interaction between the type of insecticide and clone, bl (k = 1, 2, 3) is the
random block effect, assuming bl N 0, σ 2block , IC(b)ij(l ) is the random effect of
the interaction between the insecticide and clone within blocks, assuming
insecticide × cloneðblockÞijðkÞ N 0, σ 2insecticide × clone × block , τl (l = 1, 2, 3) is the
fixed effect due to measurement time, and (Cτ)il and (ICτ)ijkl are the fixed effects
due to interaction.
9.3
Effect of Insecticides on Aphid Growth
383
Table 9.5 Effect of insecticides (C = control, R = resistant, S = susceptible) on aphid growth
Block
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
4
4
4
4
4
4
Cage
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
2
3
4
5
6
Insecticide
Control
Control
D
D
H
H
P
P
Control
Control
D
D
H
H
P
P
Control
Control
D
D
H
H
P
P
Control
Control
D
D
H
H
Clone
R
S
R
S
R
S
R
S
R
S
R
S
R
S
R
S
R
S
R
S
R
S
R
S
R
S
R
S
R
S
Dia1
60
127
64
110
118
71
66
40
54
58
76
48
130
76
93
49
94
26
121
78
73
54
25
36
75
86
69
122
185
47
Dia2
111
131
30
27
75
10
69
25
152
130
60
22
113
76
77
0
175
33
73
23
74
27
10
22
134
57
32
66
88
23
Dia6
220
220
27
35
121
111
62
19
156
362
110
110
101
85
185
8
292
52
60
1
56
49
32
1
238
194
12
20
251
116
Link function: log(λijkl) = ηijk is the link function that relates the linear predictor to
the mean (λijkl).
The following SAS program adjusts the GLMM with a Poisson distribution on
repeated measures.
proc glimmix nobound method=laplace;
class ID Block Insecticide Cage Clone time;
model y = Insecticide|clone|time/dist=poi;
random intercept Insecticide*Clon/subject=block;
lsmeans Insecticide|Clon|time/lines ilink;
run;quit;
384
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.6 Results of the analysis of variance in the Poisson GLMM
(a) Fit statistics
Fit statistics
CS
-2 Log likelihood
1125.54
AIC (smaller is better)
1177.54
AICC (smaller is better)
1202.17
BIC (smaller is better)
1161.58
CAIC (smaller is better)
1187.58
HQIC (smaller is better)
1142.52
(b) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
AR(1)
1113.19
1165.19
1189.82
1149.24
1175.24
1130.18
Toep(1)
1127.25
1177.25
1199.67
1161.91
1186.91
1143.59
UN
No converge
1006.38
484.48
5.77
Before fitting the GLMM, we compare the estimates of covariance structures with
a Poisson distribution assumed in the response variable. According to the fit statistics, the covariance structure that best models the data is the autoregressive type of
order 1 (AR(1)). The value of the fit statistic of the conditional distribution
Pearson′s chi - square/DF = 5.77 indicates that there is an extra variation (aka
overdispersion) and that the Poisson distribution does not adequately fit the data
(Table 9.6).
Since there is overdispersion in the data, a highly recommended alternative is to
find another suitable (or more appropriate) distribution for this dataset. In this case,
the linear predictor will be the same, although now, a negative binomial distribution
will be assumed in the response variable. That is,
yijkl jbl , insecticide × cloneðblockÞijðlÞ Negative Binomial λijkl , ϕ
This negative binomial model arises by assuming that the conditional distribution
of observations given random blocks and Insecticide*clone(block)ij(l )) is as follows:
yijkljbl, insecticide*clone(block)ij(l ) ~ Poisson(λijkl), where λijkl Gamma ϕ1 , ϕ .
The result of the new distribution of yijkljbl, insecticide × clone(Block)ij(l ) is a
negative binomial (Negative binomial (λijkl, ϕ)). The link function is log(λijkl) = ηijkl.
The following SAS code fits the GLMM with a negative binomial distribution.
proc glimmix nobound method=laplace;
class ID Block Insecticide Cage Clone time;
model y = Insecticide|clone|time/dist=negbi;
random intercept Insecticide*Clone/subject=block;
lsmeans Insecticide|Clone|time/lines ilink;
run;
9.3
Effect of Insecticides on Aphid Growth
385
Table 9.7 Fit statistics
(a) Fit statistics
-2 Log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
(b) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
Table 9.8 Estimated variance components and tests of
fixed effects
(a) Covariance parameter estimates
Cov Parm
Subject
Estimate
Variance
Block
0.06138
AR(1)
Block
-0.7143
Scale
0.1654
(b) Type III tests of fixed effects
Num
Den
Effect
DF
DF
Insecticide
3
19
Clone
1
19
Insecticide*clone
3
19
Time
2
44
Insecticide*time
6
44
Clone*time
2
44
Insecticide*clone*time 6
44
878.02
932.02
956.41
915.45
942.45
895.66
841.84
72.47
0.81
Standard error
0.03429
0.1710
0.03575
Fvalue
23.04
8.60
2.25
6.08
7.93
3.90
2.15
Pr > F
<0.0001
0.0086
0.1161
0.0047
<0.0001
0.0275
0.0663
Part of the results is shown in Table 9.7. The values of the fit statistics, assuming a
negative binomial distribution of the data, are shown in part (a), and the value of the
conditional statistic is observed in part (b) (Pearson′s chi - square/DF = 0.81). This
indicates that overdispersion has been eliminated from the data, and, so, the negative
binomial distribution adequately models the response variable.
The estimated variance components are shown in part (a) of Table 9.8, under an
AR(1) covariance structure. The estimates of the variance components of blocks, the
interaction between the insecticide and clone within blocks, and the scale parameter
are σ 2block = 0:06613, σ 2insecticide × cloneðblockÞ = - 0:7575, and ϕ = 0:1584, respectively.
The fixed III type effects tests (part (b)) indicate that there is a significant effect of
insecticide type (P < 0.0001), clone (P = 0.0387), measurement time (P = 0.0137),
and interactions insecticide x measurement time (P < 0.0001) and clone x measurement time (P = 0.0259) on the average number of aphids. The interaction insecticide
x clone x measurement time is close to significance (P < 0.0663).
386
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.9 Estimates of insecticide least squares (LS) means on the model scale (Estimate) and the
data scale (Mean)
Insecticide
C
D
H
P
Estimate
4.7344
3.9647
4.3733
3.4892
Standard
error
0.1478
0.1547
0.1561
0.1753
DF
19
19
19
19
tvalue
32.03
25.62
28.02
19.90
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
Mean
113.79
52.7043
79.3010
32.7588
Standard error
mean
16.8211
8.1553
12.3753
5.7432
Table 9.10 Clone least squares means on the model scale (Estimate) and the data scale (Mean)
Clone
R
S
Estimate
4.4332
3.8476
Standard error
0.1320
0.1890
DF
19
19
t-value
33.58
20.36
Pr > |t|
<0.0001
<0.0001
Mean
84.1990
46.8785
Standard error mean
11.1158
8.8586
Table 9.11 Insecticide*clone least squares means on the model scale (Estimate) and the data scale
(Mean)
Insecticide
C
C
D
D
H
H
P
P
Clone
R
S
R
S
R
S
R
S
Estimate
4.8836
4.5852
4.0521
3.8773
4.7106
4.0359
4.0866
2.8918
Standard
error
0.1529
0.2479
0.1886
0.2337
0.1997
0.2244
0.2322
0.2534
DF
19
19
19
19
19
19
19
19
tvalue
31.93
18.49
21.49
16.59
23.59
17.98
17.60
11.41
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
132.10
98.0186
57.5153
48.2958
111.11
56.5964
59.5346
18.0255
Standard
error mean
20.2032
24.3008
10.8459
11.2858
22.1870
12.7026
13.8263
4.5675
Table 9.12 Time least squares means on the model scale (Estimate) and the data scale (Mean)
Time
1
2
3
Estimate
4.2730
3.9108
4.2373
Standard error
0.1434
0.1454
0.1457
DF
44
44
44
t-value
29.79
26.90
29.09
Pr > |t|
<0.0001
<0.0001
<0.0001
Mean
71.7375
49.9372
69.2231
Standard error mean
10.2905
7.2603
10.0830
The linear predictors and estimated means of the factors and interaction are under
the “Estimate” and “Mean” columns, respectively. Average number of aphids for
insecticide, clone and time are given below:
For insecticide type (Table 9.9):
For clone (Table 9.10):
For the interaction insecticide*clone (Table 9.11):
For measurement time (Table 9.12):
For the interaction insecticide*time (Table 9.13):
For the interaction clone*time (Table 9.14):
For the interaction insecticide*clone*time (Table 9.15):
9.4
Manufacture of Livestock Feed
387
Table 9.13 Insecticide*time least squares means on the model scale (Estimate) and the data scale
(Mean)
C
C
C
D
D
D
H
H
H
P
P
P
Time
1
2
3
1
2
3
1
2
3
1
2
3
Estimate
4.2381
4.6631
5.3019
4.4854
3.7061
3.7026
4.4718
3.9790
4.6689
3.8967
3.2949
3.2759
Standard
error
0.1930
0.1913
0.1898
0.1965
0.2014
0.2035
0.1978
0.2016
0.1977
0.2241
0.2357
0.2399
DF
44
44
44
44
44
44
44
44
44
44
44
44
tvalue
21.95
24.38
27.94
22.83
18.40
18.20
22.60
19.73
23.62
17.39
13.98
13.65
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
69.2781
105.96
200.71
88.7111
40.6940
40.5537
87.5164
53.4625
106.59
49.2403
26.9755
26.4664
Standard
error mean
13.3733
20.2696
38.0898
17.4275
8.1959
8.2517
17.3133
10.7804
21.0694
11.0358
6.3583
6.3502
Table 9.14 Clone*time least squares means on the model scale (Estimate) and the data scale
(Mean)
Clone
R
R
R
S
S
S
9.4
Time
1
2
3
1
2
3
Estimate
4.3839
4.2826
4.6331
4.1621
3.5390
3.8416
Standard
error
0.1595
0.1605
0.1601
0.2092
0.2131
0.2144
DF
44
44
44
44
44
44
tvalue
27.49
26.68
28.94
19.90
16.60
17.91
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
80.1482
72.4270
102.83
64.2093
34.4308
46.5989
Standard error
mean
12.7828
11.6256
16.4644
13.4323
7.3387
9.9931
Manufacture of Livestock Feed
In this experiment, two types of pelleted feed were manufactured using different
amounts of whole sorghum. Using the whole grain resulted in one feed with a high
pellet durability index (PDI) and one with a low PDI. The researcher was interested
in how much impact this difference in PDI would have on the amount of intact and
pelleted feed distributed to the different positions along the feeding line. The line
was fed four times with the high PDI feed and four times with the low PDI feed.
After each run, the total weight of the feed in each of the 12 identified trays was
measured. The feed was then sieved into each tray, and the crushed fine granules
were weighed in the feed line. The response of interest was the ratio (proportion)
between the weight of fine granules and the total weight of the feed for each tray. The
data for this experiment are in the Appendix (Data: Feeding line experiment).
The experimental design used in this study was a split plot in a randomized
completely design. There were 2 fixed factors, feed with 2 levels (high PDI feed
(H) and low PDI feed (L)), and a tray with 12 levels (1, 2, 3, ..., 12 locations along
Insecticide
C
C
C
C
C
C
D
D
D
D
D
D
H
H
H
H
H
H
P
P
P
P
P
P
Clone
R
R
R
S
S
S
R
R
R
S
S
S
R
R
R
S
S
S
R
R
R
S
S
S
Time
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
Estimate
4.2592
4.9631
5.4284
4.2170
4.3630
5.1754
4.4161
3.8625
3.8775
4.5546
3.5497
3.5277
4.8146
4.4776
4.8395
4.1291
3.4803
4.4984
4.0456
3.8271
4.3870
3.7479
2.7627
2.1648
Standard error
0.2321
0.2280
0.2269
0.3044
0.3031
0.2999
0.2538
0.2587
0.2623
0.2908
0.2994
0.3026
0.2622
0.2639
0.2644
0.2839
0.2935
0.2823
0.3077
0.3117
0.3060
0.3189
0.3453
0.3634
DF
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
t-value
18.35
21.76
23.92
13.86
14.40
17.26
17.40
14.93
14.78
15.66
11.86
11.66
18.36
16.97
18.30
14.54
11.86
15.93
13.15
12.28
14.34
11.75
8.00
5.96
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Table 9.15 Insecticide*clone*time least squares means on the model scale (Estimate) and the data scale (Mean)
Mean
70.7546
143.04
227.78
67.8323
78.4950
176.87
82.7767
47.5825
48.3054
95.0710
34.8028
34.0460
123.30
88.0246
126.40
62.1193
32.4709
89.8763
57.1426
45.9304
80.3983
42.4308
15.8430
8.7125
Standard error mean
16.4256
32.6184
51.6892
20.6460
23.7891
53.0366
21.0081
12.3088
12.6694
27.6437
10.4203
10.3007
32.3313
23.2294
33.4231
17.6362
9.5292
25.3740
17.5807
14.3176
24.5990
13.5319
5.4700
3.1659
388
9
Generalized Linear Mixed Models for Repeated Measurements
9.4
Manufacture of Livestock Feed
Table 9.16 Results of the
analysis of variance of the
experiment
Sources of variation
Feeding
Feeding (running)
Tray
Feeding*tray
Feeding (tray*tray)
Total
389
Degrees of freedom
(2 - 1) = 1
2(4 - 1) = 6
(12 - 1) = 11
(2 - 1)(12 - 1) = 11
2(12 - 1)(4 - 1) = 66
a × b × r - 1 = 95
the feed line). Different run levels (1, 2, 3, 4 runs in the feed line) may influence the
inference of this experiment, so it is advisable to analyze which variance structure is
suitable for this analysis.
The ANOVA table (Table 9.16) with degrees of freedom for this experiment is
shown below.
The researcher aims to draw conclusions about the destructiveness in the feed line
with two types of feed, high PDI and low PDI. The following GLMM is used to
describe the experiment:
yijk = μ þ αi þ αðr Þik þ βj þ ðαβÞij þ εijk
where yijk is the proportion observed in the run k (k = 1, 2, 3, 4), tray
j ( j = 1, 2, . . ., 12), and in feed i (i = 1, 2); μ is the overall mean; αi is the fixed
effect of feed i; α(r)ik is the random effect of the ith feed within the kth run, assuming
αðr Þik N 0, σ 2αr ; βj is the fixed effect due to the jth tray; (αβ)ij is the effect of the
interaction between the ith feed and the jth tray; and εijk is the experimental error.
The components of the conditional GLMM assuming that the response variable
follows a beta distribution are listed below:
The distribution of the response variable is given by yikj j α(r)ik~Beta(μ + αi +
α(r)ik + βj + (αβ)ij, ϕ) whose linear predictor is ηijk = μ + αi + α(r)ik + βj + (αβ)ij with
π
link function logit 1 -ijkπ ijk = ηijk . The following GLIMMIX syntax fits a GLMM
with a beta distribution.
proc glimmix method=laplace;
class tray feed run;
model ratio = feed|tray/dist=beta;
random intercept/subject=feed(run) type=toep(1);
lsmeans feed|tray/lines ilink;
run;
Part of the output is shown below. Four covariance structures (“CS,” “AR(1),”
“Toep(1),” and “UN”) were tested to see which one best fits the response variable.
Of these covariance structures, “Toep(1)” produced the best fit statistics (part (a),
Table 9.17).
Another important result that gives the guideline to continue with the analysis is
the conditional distribution statistic (Pearson′s chi - square/DF = 0.96), whose
390
9
Table 9.17 Results of the
analysis of variance
Generalized Linear Mixed Models for Repeated Measurements
(a) Fit statistics
-2 Log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
(b) Fit statistics for conditional distribution
-2 Log L (ratio | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Feed
1
6
1071.19
Tray
11
65
18.03
Tray*feed
11
65
1.83
-429.84
-377.84
-357.19
-375.78
-349.78
-391.77
-453.24
91.40
0.96
Pr > F
<0.0001
<0.0001
0.0660
Table 9.18 Feed least squares means on the model scale (Estimate) and the data scale (Mean)
Feed
H
L
Estimate
-2.0009
1.3832
Standard error
0.07409
0.07208
DF
6
6
t-value
-27.01
19.19
Pr > |t|
<0.0001
<0.0001
Mean
0.1191
0.7995
Standard error mean
0.007773
0.01155
value indicates that the beta model adequately fits the data, whereas the fixed effects
tests (part (c)) indicate that there is a statistically significant effect of feeding type
(P = 0.0001) and tray (P = 0.0001).
The linear predictors and estimated probabilities of the factors and interaction are
listed under the “Estimate” and “Mean” columns of the following tables,
respectively.
For the feeding line (Table 9.18):
For the tray (Table 9.19):
For the interaction feeding*tray (Table 9.20):
9.5
Characterization of Spatial and Temporal Variations
in Fecal Coliform Density
During a 1-month period (June 1981), 30 river water samples were collected from
the channel at 3 stations, A, B, and C (downstream to upstream) on 5 randomly
selected days at 9:00 a.m. and 3:00 p.m. (1 sample per station per hour per day). Each
sample was analyzed for fecal coliform by method FC-96. The data from this
experiment are shown in Table 9.21.
9.5
Characterization of Spatial and Temporal Variations in Fecal Coliform Density
391
Table 9.19 Tray least squares means on the model scale (Estimate) and the data scale (Mean)
Tray
01
02
03
04
05
06
07
08
09
10
11
12
Estimate
-0.5652
-0.6607
-0.6950
-0.2958
-0.3773
-0.2947
-0.3520
-0.2992
-0.1314
-0.3935
0.1571
0.2014
Standard error
0.08182
0.08531
0.08822
0.08100
0.08212
0.08057
0.08165
0.07939
0.07670
0.08096
0.07860
0.07949
DF
65
65
65
65
65
65
65
65
65
65
65
65
t-value
-6.91
-7.74
-7.88
-3.65
-4.59
-3.66
-4.31
-3.77
-1.71
-4.86
2.00
2.53
Pr > |t|
<0.0001
<0.0001
<0.0001
0.0005
<0.0001
0.0005
<0.0001
0.0004
0.0916
<0.0001
0.0499
0.0137
Mean
0.3623
0.3406
0.3329
0.4266
0.4068
0.4268
0.4129
0.4258
0.4672
0.4029
0.5392
0.5502
Standard error mean
0.01891
0.01916
0.01959
0.01981
0.01982
0.01971
0.01979
0.01941
0.01909
0.01948
0.01953
0.01967
Table 9.20 Tray*feed least squares means on the model scale (Estimate) and the data scale (Mean)
Tray
01
01
02
02
03
03
04
04
05
05
06
06
07
07
08
08
09
09
10
10
11
11
12
12
Feed
H
L
H
L
H
L
H
L
H
L
H
L
H
L
H
L
H
L
H
L
H
L
H
L
Estimate
-2.2408
1.1104
-2.4581
1.1367
-2.4724
1.0823
-2.0307
1.4391
-2.1481
1.3935
-2.0087
1.4192
-2.1026
1.3987
-1.9310
1.3325
-1.6240
1.3613
-2.0863
1.2994
-1.4559
1.7701
-1.4519
1.8548
Standard
error
0.1284
0.1015
0.1369
0.1018
0.1375
0.1105
0.1217
0.1070
0.1254
0.1061
0.1208
0.1066
0.1242
0.1061
0.1192
0.1050
0.1113
0.1056
0.1238
0.1044
0.1075
0.1148
0.1076
0.1171
DF
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
65
t-value
-17.46
10.94
-17.95
11.17
-17.98
9.79
-16.69
13.45
-17.13
13.13
-16.62
13.31
-16.93
13.18
-16.20
12.69
-14.59
12.89
-16.85
12.45
-13.55
15.42
-13.50
15.83
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
0.09614
0.7522
0.07885
0.7571
0.07782
0.7469
0.1160
0.8083
0.1045
0.8011
0.1183
0.8052
0.1088
0.8020
0.1266
0.7913
0.1647
0.7960
0.1104
0.7857
0.1891
0.8545
0.1897
0.8647
Standard error
mean
0.01116
0.01892
0.009946
0.01872
0.009869
0.02089
0.01248
0.01658
0.01174
0.01690
0.01260
0.01673
0.01204
0.01685
0.01318
0.01734
0.01531
0.01715
0.01216
0.01757
0.01648
0.01427
0.01653
0.01371
392
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.21 Variation in fecal coliform densities of the river water samples from three sampling
stations on five sampling days at 9:00 a.m. (TM = 1) and 3:00 p.m. (TM = 2)
Sampling date
18 May
18 May
18 May
18 May
18 May
18 May
26 May
26 May
26 May
26 May
26 May
26 May
29 May
29 May
29 May
29 May
29 May
29 May
1 June
1 June
1 June
1 June
1 June
1 June
5 June
5 June
5 June
5 June
5 June
5 June
TM
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
9:00 a.m.
3:00 p.m.
Site
A
A
B
B
C
C
A
A
B
B
C
C
A
A
B
B
C
C
A
A
B
B
C
C
A
A
B
B
C
C
No. of coliforms per milliliter
648
798
517
702
532
55
1421
1388
1883
1855
1724
1769
1523
759
1361
603
2004
541
1987
1056
1796
1579
1221
1223
870
1099
920
951
926
887
To assess the relative magnitudes of sources of variation due to time, site, and
subsampling on the number of coliforms per milliliter (yijk), an analysis of variance
using a GLMM with a Poisson response was performed, as described below:
We denote yijk as the number of colonies per milliliter, whose conditional
distribution is given by yijkjsampling(site)ik ~ Poisson (λijk) with the linear predictor
ηijk defined by
ηijk = θ þ sitei þ samplingðsiteÞik þ timej þ ðsite × timeÞij
9.5
Characterization of Spatial and Temporal Variations in Fecal Coliform Density
393
Table 9.22 Results of the analysis of variance
(a) Estructuras de covarianza
Fit statistics
Toep(1)
CS
-2 Log likelihood
2022.64
2022.64
AIC (smaller is better)
2054.64
2056.64
AICC (smaller is better)
2096.49
2107.64
BIC (smaller is better)
2051.31
2053.10
CAIC (smaller is better)
2067.31
2070.10
HQIC (smaller is better)
2041.31
2042.47
(b) Fit statistics for conditional distribution
-2 Log L (ufc | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
Site
2
3
T
4
12
T*site
8
12
AR(1)
2022.64
2056.64
2107.64
2053.10
2070.10
2042.47
UN
2022.64
2054.64
2096.49
2051.31
2067.31
2041.31
1989.36
1632.28
54.41
F-value
1.41
956.04
82.44
Pr > F
0.3700
<0.0001
<0.0001
ði = 1, 2, 3; j = 1, 2, 3, 4, 5; k = 1, 2Þ
where ηijk is the linear predictor that relates the linear function to the mean, θ is the
intercept, sitei is the fixed effect due to the sampling site i, sampling(site)ik is the
random effect due to the sampling time nested within the site, assuming
samplingðsiteÞik N 0, σ 2samplingðsiteÞ , timej is the fixed effect due to sampling
date, and (site × time)ij is the effect of the interaction between the site and sampling
date. The link function for this model is log(λijk) = ηijk.
The following GLIMMIX syntax fits a GLMM with a Poisson response.
proc glimmix data=ufc nobound method=laplace;
class T TM Site ;
model ufc = Site|T/dist=Poisson link=log;
random intercept/subject=TM(Site) type=toep(1);
lsmeans Site|T/lines ilink;
run;
Part of the results is summarized in Table 9.22. To determine which covariance
structure best models the response variable, four types were tested (part (a)), all of
which produced very similar results. Because of these results, the “Toep(1)” covariance structure was chosen. From this, the fit statistics were obtained, and the value of
the conditional distribution statistic is Pearson′s chi - square/DF = 54.41. This
value indicates that there is a strong overdispersion in the dataset. Therefore, it is
important to look for an alternative distribution that solves this problem.
The hypothesis tests in part (c) indicate that there is a significant difference in the
date of sampling (P = 0.0001) as well as in the interaction between the site and date
394
Table 9.23 Fit statistics
under the negative binomial
distribution
9
Generalized Linear Mixed Models for Repeated Measurements
(a) Fit statistics
-2 Log likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
(b) Fit statistics for conditional distribution
-2 Log L(ufc | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
436.98
470.98
521.98
467.44
484.44
456.81
432.83
22.66
0.76
of sampling (P = 0.0001). That is, the concentration of fecal coliform units per
milliliter is affected by the date of data collection. However, we observed that there
is an excessive dispersion in the data. One way to check for and deal with
overdispersion is to run a quasi-Poisson model, which, during the fitting process,
adds an additional dispersion parameter to account for that additional variance.
Another option is to look for a distribution that adequately fits the data; in this
case, the negative binomial distribution is a good alternative.
Next, we will implement the analysis assuming that the response variable is
distributed under a negative binomial distribution. This means that the distribution
of yijk (number of colonies per militro) is given by yijk j smapling(site)ik~Negative
Binomial (λijk, ϕ), where ϕ is the scale parameter. However, the linear predictor ηijk
and the link function remain unchanged.
The following GLIMMIX commands fit a GLMM with a negative binomial
distribution.
proc glimmix data=ufc nobound method=laplace;
class T TM Site;
model ufc = Site|T/dist=negbin;
random intercept/subject=TM(Site)/type=Toep(1);
lsmeans Site|T/lines ilink;
run;
Part of the output of the above program is shown below. The values of the fit
statistics under the negative binomial distribution (part (a) of Table 9.23) are much
smaller compared to those obtained assuming the Poisson model, indicating that the
negative binomial distribution adequately fits the response variable. Furthermore,
the value of the conditional distribution statistic indicates that the negative binomial
distribution is a good distribution for these data (Pearson′s chi - square/DF = 0.76).
refers to how many times the
This parameter Pearson0 s chi - square
DF = 0:76
variance is larger than the mean. Since this value is less than 1 (part (b)), the
conditional variance is actually smaller than the conditional mean, indicating that
overdispersion has been removed in the fitting of the data. Another direct effect
9.6
Log-Normal Distribution
Table 9.24 Type III fixed
effects tests
395
Effect
Site
T
T*site
Num DF
2
4
8
Den DF
3
12
12
F-value
0.78
11.57
1.13
Pr > F
0.5346
0.0004
0.4096
Table 9.25 Means and standard errors on the model scale (Estimate) and on the data scale (Mean)
of the sampling site data
Site
A
B
C
Estimate
7.0195
7.0243
6.8237
Standard error
0.1270
0.1271
0.1305
DF
3
3
3
t-value
55.25
55.28
52.30
Pr > |t|
<0.0001
<0.0001
<0.0001
Mean
1118.25
1123.57
919.40
Standard error mean
142.06
142.78
119.95
Table 9.26 Means and standard errors of measurement time on the model scale (Estimate) and the
data scale (Mean)
T
1
2
3
4
5
Estimate
6.2084
7.4212
7.0074
7.2910
6.8513
Standard error
0.1467
0.1420
0.1455
0.1418
0.1422
DF
12
12
12
12
12
t-value
42.32
52.27
48.16
51.42
48.19
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
496.91
1670.97
1104.74
1466.97
945.09
Standard error mean
72.8990
237.23
160.75
208.00
134.35
observed when there is no overdispersion is the F-values of the fixed effects tests
(Table 9.24). In this case, the date on which the samples were collected was
significant but not the interaction between the two factors, as the case when the
data were fitted using the Poisson GLMM.
The linear predictors and estimated probabilities of the main effects and the
interaction between both factors are under the columns “Estimate” and “Mean,”
respectively. The sampling site averages are presented below (Table 9.25).
The averages by sampling date are listed below (Table 9.26).
The means of the interaction site × sampling date are shown below (Table 9.27).
9.6
Log-Normal Distribution
Positively skewed distributions are highly common, especially when modeling
biological data. Data often have a lower bound, usually 0 or the detection limit,
but have no restriction on the upper bound. Therefore, when the data are below the
median, no observation can be further away than the lower bound; however, when
the data are above the median, there may be values that are many times further away,
giving a positively skewed distribution. These skewed distributions can often be
approximated by a log-normal distribution (Limpert et al. 2001).
396
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.27 Means and standard errors for the interaction T*site on the model scale (Estimate) and
the data scale (Mean)
Site
A
B
C
A
B
C
A
B
C
A
B
C
A
B
C
T
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
Estimate
6.5905
6.4197
5.6151
7.2508
7.5367
7.4761
7.0336
6.8855
7.1030
7.3224
7.4329
7.1176
6.9003
6.8467
6.8069
Standard
error
0.2463
0.2466
0.2772
0.2452
0.2451
0.2463
0.2461
0.2465
0.2586
0.2458
0.2450
0.2463
0.2460
0.2458
0.2453
DF
tvalue
26.76
26.03
20.26
29.57
30.75
30.36
28.58
27.94
27.47
29.79
30.33
28.90
28.04
27.85
27.75
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
728.17
613.79
274.53
1409.17
1875.54
1765.30
1134.09
978.01
1215.59
1513.87
1690.62
1233.47
992.56
940.73
904.07
Standard error
mean
179.35
151.38
76.1038
345.59
459.64
434.71
279.14
241.05
314.37
372.07
414.28
303.81
244.21
231.25
221.80
Fig. 9.1 Density function of the log-normal distribution with parameters 1 and 0.6
A log-normal distribution is characterized by having only positive nonzero
values, positive skewness, a nonconstant variance that is proportional to the square
of the mean value, and a normally distributed natural logarithm. The probability
density function for a log-normal distribution has an asymmetric appearance, with a
larger amount of data below the expected value and a thinner right tail with higher
values. Figure 9.1 shows the positive skewness of a log-normal distribution with
mean 1 and standard deviation 0.6.
9.6
Log-Normal Distribution
9.6.1
397
Emission of Nitrous Oxide (N2O) in Beef Cattle Manure
with Different Percentages of Crude Protein in the Diet
The experiment was conducted between January and February 2017 at the Colegio
de Postgraduados Campus Córdoba located in Amatlán de los Reyes, Veracruz,
México. The genetic material used were four 5–6-month-old males of the Criollo
lechero tropical (CLT) breed, randomly distributed in individual pens of
4.8 × 2.1 m2, each one with 75% shade, a cup drinker, and a drawer-type feeder.
To ensure the required crude protein percentages for each treatment, the following
diets (treatments 1–4) were developed: Trt1 (12% crude protein), Trt2 (14% crude
protein), Trt3 (16% crude protein), and Trt4 (commercial feed with 16% crude
protein). Each animal randomly received the four treatments in different periods.
Each treatment was applied for 11 days, of which the first 7 were considered
adaptation days and the following 4 days were used for the measurement of gases
in the daily accumulated excreta. The experiment had a total duration of 44 days. The
data from this experiment are tabulated in the Appendix (Data: Nitrous oxide
emission). The N2O gas fluxes in ppm were calculated from a linear or nonlinear
increase of the concentrations inside the static chambers over time, and these fluxes
were converted to micrograms of N2O–N per m2 per hour ( y); for more details, see
the study by Nadia Hernández-Tapia et al., (2019). The statistical model used in this
study was an analysis of covariance model in a randomized complete block design
with repeated measures, as described below.
yijk = μ þ τi þ animalj þ timek þ ðτ × timeÞik þ βi xij - x þ εijk
where yijk is the flux of N2O–N (μg m-2 h-1); μ is the overall mean; τi is the fixed
effect due to treatment i (i = 1, 2, 3, 4); animalj is the random effect due to animal
j ( j = 1, 2, 3, 4), assuming animalj~N(0, σ 2animal); timek is the fixed effect of time
k (k = 1, 2, 3, 4, 5) at the time of measurement; (τ × time)ik is the effect of the
interaction between τi and timek, βi is the coefficient of linear regression of the
covariate xij in treatment i and time j, where xij can be the pH, humidity (HE),
temperature (TE) in the manure, maximum temperature (TMaxA), minimum temperature (TMinA), maximum humidity (HMaxA), minimum humidity (HMinA), or
initial weight (kilograms) at the start of a treatment; x is the mean of the covariate in
question; and εijk is the non-normal experimental error.
The linear predictor ηijk for N2O–N is ηijk = μ þ τi þ animalj þ timek þ
ðτ timeÞik þ βi xij - x . The response variable yijk has a conditional log-normal
distribution with a mean μijk and variance eσ - 1 :e2μþσ , that is, yijkjanimalj ~ Log
2
normal (μijk,
2
eσ - 1 :e2μþσ ); the rest of the parameters have already been
described above.
2
2
398
9
Generalized Linear Mixed Models for Repeated Measurements
The following GLIMMIX syntax adjusts a GLMM with a log-normal response:
proc glimmix data=co2 nobound method=laplace;
class animal trt time;
model flox =trt|time xbar/dist=lognormal;
random intercept /subject=animal type=cs;
lsmeans trt|time/lines ilink;
run;
Although most of the commands have already been described in previous
chapters, in this chapter, we average the TMinA covariate “xbar.” Part of the output
is shown below.
The gas emissions from cattle manure, regardless of the treatment applied, are
influenced by several factors (covariates) that the researcher cannot control, which
have a significant effect on the estimation of means and experimental error. Both are
linearly related to the response variable. Covariates such as pH, humidity, and
temperature of the excreta, as well as the temperature and humidity (maximum
and minimum) of the environment, influence the dynamics of gas emission. These
covariates were considered and analyzed in the covariance model to adjust the
estimated means of the N2O–N flux. Based on the fit statistics obtained from the
proposed models (Table 9.28), the model that best explains the variability of the
N2O–N flux is model 5 because this model provides the lowest values in AIC, AICC,
BIC, and MSE (Mean Square Error). Therefore, the model that provides the best fit
or explains the most variability in the N2O–N flux is the one that includes the
minimum environment temperature.
The conditional fit statistics (part (a)) and the estimated variance components
(part (b)) are shown in Table 9.29. The type III fixed effects tests (part (c)) indicate
that there is a significant effect of Trt (P = 0.0008), time (P = 0.0288), the
interaction Trt × time (P = 0.0140), the covariate Tmin (P = 0.0079), and the
interaction Tmin × Trt (P = 0.038).
The average N2O–N emissions between Trt1 (12% CP: Crude Protein) and Trt2
(14% CP) were statistically different from each other. Treatment 1 emitted the
highest N2O–N flux despite being the treatment with the lowest percentage of CP
(Table 9.30).
9.7
Effect of a Chemical Salt on the Percentage Inhibition
of the Fusarium sp.
In order to observe the tolerance of the fungus Fusarium sp. to different concentrations of a chemical salt, a bioassay was implemented to evaluate the percentage of
inhibition of the fungus. This bioassay consisted of placing a nutritive culture
medium in Petri dishes for the fungal development in which different concentrations
of the salt in ppm were added (0, 500, 1000, and 2000, ). Mycelium growth was
measured during 6 days, and the percentage of inhibition of Fusarium sp. growth
was calculated. Part of the data is shown below, and the complete base is in the
Appendix (Data: Percentage inhibition).
417.62
443.88
416.78
5. Y ijk = μ þ τi þ animalj þ timek þ ðτ × timeÞik þ β TMinAij - TMinA þ εijk
6. Y ijk = μ þ τi þ animalj þ timek þ ðτ × tiempoÞik þ β HMaxAij - HMaxA þ εijk
7. Y ijk = μ þ τi þ animalj þ timek þ ðτ × timeÞik þ β HMinAij - HMinA þ εijk
450.66
4. Y ijk = μ þ τi þ animalj þ timek þ ðτ × timeÞik þ β TMaxAij - TMaxA þ εijk
428.32
455.42
459.14
462.20
454.28
429.98
402.05
429.15
391.84
435.93
428.01
403.71
418.44
442.74
2. Y ijk = μ þ τi þ animalj þ timek þ ðτ × timeÞik þ β HEij - HE þ εijk
BIC
407.67
Fit statistics (N2O–N)
AIC
AICC
422.40
433.94
3. Y ijk = μ þ τi þ animalj þ timek þ ðτ × tiemeÞik þ β TE ij - TE þ εijk
1. Y ijk = μ þ τi þ animalj þ timek þ ðτ × timeÞik þ β pHχ ij - pH þ εijk
Model with the specific covariate
Table 9.28 Fit statistics in the different models proposed
0.99
1.18
0.76
1.25
1.16
0.97
CME
1.09
9.7
Effect of a Chemical Salt on the Percentage Inhibition of the Fusarium sp.
399
400
Table 9.29 Conditional fit
statistics, variance components, and type III fixed effect
tests
Table 9.30 Mean and standard error of N flux2 O (μg of
N2O-N m-2 h-1) of the different treatments under study
Bio
1
1
⋮
1
1
⋮
1
1
1
1
⋮
1
1
⋮
1
1
1
Generalized Linear Mixed Models for Repeated Measurements
9
Day
1
1
⋮
2
2
⋮
3
3
4
4
⋮
5
5
⋮
6
6
6
Conc
0
0
⋮
0
0
⋮
0
0
0
500
⋮
0
0
⋮
1000
1000
2000
(a) Fit statistics for conditional distribution
-2 Log L (F | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Subject
Estimate
Variance
A
-0.01561
CS
A
0.000767
Residual
0.7776
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
Trt
3
88
Time
4
88
Trt*time
11
88
Tmin
1
88
Tmin*Trt
3
88
Tmin*time
4
88
Tmin*Trt*time
11
88
Treatment
Trt1 (12% PC)
Trt2 (14% PC)
Trt3 (16% PC)
Trt4 (16% CP, commercial feed)
Rep
3
4
⋮
2
3
⋮
2
3
3
1
⋮
3
4
⋮
3
4
1
Y
5.263
5.263
⋮
1.935
4.516
⋮
1.234
3.703
4.672
19.626
⋮
4.065
4.065
⋮
15.862
18.62
32.413
Bio
2
2
⋮
2
2
⋮
2
2
2
2
⋮
2
2
⋮
2
2
2
Day
1
1
⋮
2
2
⋮
3
3
4
4
⋮
5
5
⋮
6
6
6
333.62
99.06
0.76
Standard error
.
.
0.09845
F-value
6.13
2.84
2.34
7.40
4.81
1.80
1.23
Pr > F
0.0008
0.0288
0.0140
0.0079
0.0038
0.1351
0.2814
N2O(μ) ± standard error
3.6442 ± 0.2213a
3.0714 ± 0.3119b
3.5706 ± 0.2974ab
3.1205 ± 0.2130ab
Conc
0
0
⋮
500
500
⋮
500
500
500
500
⋮
500
500
⋮
2000
2000
2000
Rep
2
3
⋮
2
3
⋮
3
4
2
3
⋮
1
2
⋮
1
2
3
Y
0.0016
14.285
⋮
31.506
42.465
⋮
35.042
24.786
23.123
27.927
⋮
13.253
21.285
⋮
31.197
29.173
29.848
(continued)
9.7
Effect of a Chemical Salt on the Percentage Inhibition of the Fusarium sp.
Bio
1
1
1
Day
6
6
6
Conc
2000
2000
2000
Rep
2
3
4
Y
29.655
31.724
35.172
Bio
2
Day
6
Conc
2000
401
Rep
4
Y
30.522
Following the same reasoning as in previous examples, the components of the
GLMM with beta response distribution repeated-measures for the percentage inhibition of Fusarium sp. (yijkl) are listed below:
Distributions: yijkl j ωkl, conc(ω)i(kl)~Beta(π ijkl, ϕ); i = 1, ⋯, 4; j = 1, . . ., 6; k = 1, 2;
l = 1, . . ., ri. ωkl N 0, σ 2ω , concðωÞiðklÞ N 0, σ 2concðωÞ .
Linear predictor: ηijk = θ þ conci þ ωkl þ concðωÞiðklÞ þ timej þ ðconc × timeÞij:
where ηijk is the linear predictor, θ is the intercept, conci is the fixed effect of salt
concentration, ωkl is the random effect of the Petri dish within the bioassay,
assuming ωkl N 0, σ 2ω , conc(ω)i(kl) is the random effect of salt concentration–
Petri dish–bioassay, assuming concðωÞiðklÞ N 0, σ 2concðωÞ , timej is the fixed effect
due to the day of measurement, and (conc × time)ij is the interaction effect of
chemical salt concentration with the day of measurement.
Link function: logit(π ijkl) = ηijkl is the link function that relates the linear predictor to
the mean (π ijkl).
The following SAS program adjusts the beta GLMM with repeated measures.
proc glimmix data=inhibition method=laplace nobound;
class Bio Day Conc Rep;
model pct = Con|Day/dist=beta link=logit;
random intercept/subject=con(bio) type=cs;
lsmeans Con|Day/lines ilink;
run;
Before fitting the generalized linear mixed model, we compare the estimates of
the covariance structures with the beta distribution in the response variable
(Table 9.31 part (a)). According to the fit statistics, the covariance structures that
best fit the data are the Toeplitz type (Toep(1)) and unstructured (UN).
Having defined the covariance structure, in this case, Toeplitz of order 1, we
present part of the results of the data fit (Table 9.31 part (b)). The fit statistic
Pearson′s chi - square/DF = 1.07 indicates that there is no overdispersion and
that the beta distribution fits the data adequately. The estimated variance component,
under Toeplitz (1), of the concentration–repetition bioassay is σ 2conðωÞ = 0:00285 and
the scale parameter ϕ = 52:281 (c).
402
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.31 Fit statistics for the conditional distribution and variance components
(a) Fit statistics
CS
-2 Log likelihood
-523.69
AIC (smaller is better)
-469.69
AICC (smaller is better)
-458.73
BIC (smaller is better)
-467.54
CAIC (smaller is better)
-440.54
HQIC (smaller is better)
-484.16
(b) Fit statistics for conditional distribution
-2 Log L (pct | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
(c) Covariance parameter estimates
Cov Parm
Subject
Variance
Con(Bio)
Scale
Table 9.32 Type III fixed
effects tests
AR(1)
-523.69
-469.69
-458.73
-467.54
-440.54
-484.16
Toep(1)
-523.69
-471.69
-461.59
-469.62
-443.62
-485.62
UN
-523.69
-471.69
-461.59
-469.62
-443.62
-485.62
-529.79
177.68
1.07
Estimate
0.002849
52.2809
Type III tests of fixed effects
Effect
Num DF
Den DF
Con
3
4
Day
5
138
Day*Con
15
138
Standard error
0.004147
5.8849
F-value
125.40
10.99
2.25
Pr > F
0.0002
<0.0001
0.0074
The fixed effects indicate that there is a highly significant effect of salt concentration (P = 0.0002), time (P = 0.0001), and the interaction concentration x time
(P = 0.0074) on the growth inhibition of Fusarium sp. (Table 9.32).
The linear predictors and estimated probabilities of the factors (Table 9.33 parts
(a) and (b)) and interaction (Table 9.34) are found under the columns “Estimate” and
“Mean,” respectively.
9.8
Carbon Dioxide (CO2) Emission as a Function of Soil
Moisture and Microbial Activity
Productive agricultural soil requires a certain level of ventilation to maintain active
plant root growth and soil microbial activity. One scientist found that soil oxygenation levels had been affected in soils fertilized with nutrient-rich sludge from a
sewage treatment plant. The level of soil aeration can be reduced by (1) the high
water content of the sludge added, through compaction with heavy machinery used
to add the sludge and, ironically, (2) the increased microbial activity that occurs
when sludge with high organic matter content is added. The objective of the research
was to determine the moisture levels at which aeration becomes a limiting factor for
9.8
Carbon Dioxide (CO2) Emission as a Function of Soil Moisture. . .
403
Table 9.33 Concentration and measurement time least square means on the model scale (Estimate)
and the data scale (Mean)
(a) Conc least squares means
Standard
Estimate
error
Con ηi:
0
-3.5438
0.1499
500
-1.0650
0.05941
1000 -0.9847
0.05895
2000 -0.4487
0.05891
(b) Day least squares means
Standard
Estimate
error
Day η:j
1
-1.6017
0.1161
2
-1.0446
0.08689
3
-1.2475
0.08794
4
-1.5668
0.1020
5
-1.7606
0.1039
6
-1.8422
0.1067
DF
4
4
4
4
t-value
-23.64
-17.93
-16.70
-7.62
Pr > |t|
<0.0001
<0.0001
<0.0001
0.0016
Mean
π i:
0.02809
0.2563
0.2720
0.3897
Standard error
mean
0.004093
0.01133
0.01167
0.01401
DF
138
138
138
138
138
138
t-value
-13.79
-12.02
-14.19
-15.36
-16.94
-17.26
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
π :j
0.1677
0.2603
0.2231
0.1727
0.1467
0.1368
Standard error
mean
0.01621
0.01673
0.01524
0.01457
0.01301
0.01260
microbial activity in the soil. The study included a control treatment (no sludge) and
three treatments using sludge as a fertilizer with different moisture contents, whose
moisture levels for the fertilized soil were 0.24, 0.26, and 0.28 kg water/kg soil.
Soil samples were randomly assigned to the four treatments in a randomized
completely design. Soil samples were placed in sealed containers and incubated
under favorable conditions for microbial activity. The soil was compacted in the
containers simulating a degree of compaction experienced in the field. Microbial
activity, measured as an increase in CO2, was used as a measure of the level of soil
oxygenation. The CO2 evolution/kilogram soil/day in each container was measured
on 2, 4, 6, and 8 days after starting of the incubation period. Microbial activity in
each soil sample was recorded as the percentage increase in CO2 produced above the
atmospheric level. The data are shown in Table 9.35.
The analysis of variance table for this experiment is shown below (Table 9.36).
Let pctijk be the percentage of CO2 emission, assuming that pctijk has a beta
distribution with a mean π ijk and scale parameter ϕ, i.e., pctijk~Beta(π ijk, ϕ). The
linear predictor ηijk that relates the mean to the link function is given by
ηijk = θ þ αi þ αðr ÞiðkÞ þ τj þ ðατÞij ; i = 1, . . . , 4, j = 1, . . . , 4, k = 1, 2, 3
where θ is the intercept, αi is the fixed effect of the treatment i, α(r)i(k) is the random
effect of treatment nested in the repetition k, assuming that αðr ÞiðkÞ N 0, σ 2αðrÞ , τj
is the fixed effect of measurement time j, and (ατ)ij is the interaction effect of
treatment with measurement time. The link function is defined by logit(π ijk) = ηijk.
The following SAS syntax fits a GLMM on repeated measures with a beta
distribution.
404
9 Generalized Linear Mixed Models for Repeated Measurements
Table 9.34 Measuring time*salt concentration interaction on the model scale (Estimate) and the
data scale (Mean)
Day*con least squares means
Day
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
Con
0
500
1000
2000
0
500
1000
2000
0
500
1000
2000
0
500
1000
2000
0
500
1000
2000
0
500
1000
2000
Estimate
ηij
-4.0127
-0.8709
-0.6848
-0.8382
-3.5743
-0.4140
-0.3519
0.1616
-2.9511
-0.9923
-0.9044
-0.1423
-3.5167
-1.2429
-1.0361
-0.4716
-3.5503
-1.4180
-1.4489
-0.6251
-3.6579
-1.4522
-1.4823
-0.7765
Standard
error
0.4083
0.1123
0.1092
0.1579
0.2957
0.1061
0.1053
0.1043
0.2944
0.1149
0.1131
0.1041
0.3558
0.1213
0.1159
0.1065
0.3579
0.1269
0.1277
0.1083
0.3691
0.1277
0.1289
0.1106
DF
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
138
t-value
-9.83
-7.76
-6.27
-5.31
-12.09
-3.90
-3.34
1.55
-10.02
-8.64
-8.00
-1.37
-9.88
-10.25
-8.94
-4.43
-9.92
-11.17
-11.34
-5.77
-9.91
-11.37
-11.50
-7.02
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
0.0001
0.0011
0.1235
<0.0001
<0.0001
<0.0001
0.1739
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Mean
π ij
0.01776
0.2951
0.3352
0.3019
0.02727
0.3980
0.4129
0.5403
0.04969
0.2705
0.2881
0.4645
0.02884
0.2239
0.2619
0.3842
0.02791
0.1950
0.1902
0.3486
0.02514
0.1897
0.1851
0.3151
Standard error
mean
0.007124
0.02335
0.02434
0.03328
0.007844
0.02543
0.02554
0.02590
0.01390
0.02266
0.02319
0.02590
0.009967
0.02108
0.02241
0.02520
0.009710
0.01992
0.01967
0.02458
0.009046
0.01963
0.01944
0.02388
proc glimmix data=co2 method=laplace;
class trt container time;
model pct = trt|time/dist=beta link=logit;
random trt/subject=container;
lsmeans trt|time /lines ilink;
run;
Part of the results is shown below. The fit statistics under different covariance
structures (Table 9.37 part (a)), such as AIC and AICC indicate that a Toeplitz-type
covariance structure of order 1 provides the best fit to the dataset of this experiment.
Table 9.38 part (a) shows the estimated variance component due to treatment x
repetition, i.e., - σ 2aðrÞ = 0:03363, and the estimated scale parameter ϕ = 790:82, and
the hypothesis test (part (b)) indicates that the treatments yielded statistically different means (P = 0.0011).
9.8
Carbon Dioxide (CO2) Emission as a Function of Soil Moisture. . .
405
Table 9.35 Repeated measurements of emissions of CO2 by bacterial activity in soil under
different moisture conditions
Moisture (kg water/kg soil)
Control
0.24
0.26
0.28
Table 9.36 Analysis of variance of an RCD with repeated
measures
Container
1
2
3
1
2
3
1
2
3
1
2
3
%CO2 evolution/kilogram soil/day
Day 2
Day 4
Day 6
0.22
0.56
0.66
0.68
0.91
1.06
0.68
0.45
0.72
2.53
2.7
2.1
2.59
1.43
1.35
0.56
1.37
1.87
0.22
0.22
0.2
0.45
0.28
1.24
0.22
0.33
0.34
0.22
0.8
0.8
0.22
0.62
0.89
0.22
0.56
0.69
Sources of variation
Treatment
Error1
Measurement time
Treatment x time
Error2
Total
Day 8
0.89
0.8
0.89
1.5
0.74
1.21
0.11
0.86
0.2
0.37
0.95
0.63
Degrees of freedom
(a - 1) = 4 - 1 = 3
a(r - 1) = 8
(t - 1) = 4 - 1 = 3
(a - 1)(t - 1) = 9
a(t - 1)(r - 1) = 4 × 3 × 2 = 24
a × t × r - 1 = 4 × 4 × 3 - 1 = 47
Table 9.37 Fit statistics of the beta GLMM under different covariance structures
(a) Fit statistics
CS
-2 Log likelihood
-433.28
AIC (smaller is better)
-395.28
AICC (smaller is better)
-368.14
BIC (smaller is better)
-412.41
CAIC (smaller is better)
-393.41
HQIC (smaller is better)
-429.71
(b) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
AR(1)
Toep(1)
-433.94
-433.28
-395.94
-397.28
-368.80
-373.69
-413.07
-413.50
-394.07
-395.50
-430.37
-429.89
CS
AR(1)
Toep(1)
-446.54 -444.41 -446.58
30.46
33.98
30.38
0.63
0.71
0.63
UN
No converge
UN
No converge
Table 9.39 shows the estimated average emissions of CO2 in tested treatments,
which showed that the treatment with moisture 0.24 kg water/kg soil favored a
higher microbial activity, whereas treatments with moisture levels 0.26 and 0.28 kg
water/kg soil showed similar microbial activity between them.
406
9
Table 9.38 Variance components and fixed effects test
Generalized Linear Mixed Models for Repeated Measurements
(a) Covariance parameter estimates
Cov Parm
Subject
Estimate
Standard error
Variance
Contenedor
0.03363
0.03153
Scale (ϕ)
790.82
190.89
(b) Type III tests of fixed effects
Effect
Num DF
Den DF
F-value
Pr > F
Trt
3
8
15.52
0.0011
Time
3
24
2.94
0.0537
Trt*time
9
24
1.29
0.2914
Table 9.39 Means and standard errors on the model scale (Estimate) and the data scale (Mean)
(a) Trt least squares means
Standard
Trt
Estimate error
C
-4.9242 0.1595
T0.24 -4.1331 0.1343
T0.26 -5.5728 0.1898
T0.28 -5.1588 0.1728
DF
8
8
8
8
t-value
-30.87
-30.79
-29.36
-29.86
Pr > |t|
<0.0001
<0.0001
<0.0001
<0.0001
Mean
0.007216
0.01578
0.003786
0.005716
Standard error
mean
0.001143
0.002085
0.000716
0.000982
0.02
0.018
0.016
% CO2
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
1
2
3
4
5
6
7
8
Time (Days)
C
T0.24
T0.26
T0.28
Fig. 9.2 CO2 emission as a measure of microbial activity
Figure 9.2 clearly shows that the treatment with moisture 0.24 kg water/kg soil
provides the best conditions for soil microbial activity, whereas the rest of the
treatments significantly affect the activity of microorganisms.
9.9
9.9
Effect of Soil Compaction and Soil Moisture on Microbial Activity
407
Effect of Soil Compaction and Soil Moisture
on Microbial Activity
A soil scientist conducted an experiment to evaluate the effects of soil compaction
and soil moisture on microbial activity. Ventilation levels may be restricted in highly
saturated or compacted soils, thus reducing microbial activity. The experiment
consisted of three levels of soil compaction (1.1, 1.4, and 1.6 mg soil/m3) and
three levels of soil moisture (0.1, 0.2, and 0.24 kg water/kg soil). The treated soil
samples were placed in sealed containers and incubated under conditions to microbial activity. The percentage increase in CO2 produced above atmospheric levels was
measured in each soil sample. The experimental design was a completely randomized design (CRD) with a 3 X 3 factorial structure of treatments. Two replicates of
the soil container units were prepared for each treatment. The evolution of CO2/kg
soil/day was measured for three successive days. The data from this experiment are
shown below in Table 9.40.
The analysis of variance table for this experiment is shown below (Table 9.41).
Let pctijk be the percentage of CO2 emission and assume that pctijk has a beta
distribution with a mean π ijk and scale parameter ϕ, i.e., pctijk~Beta(π ijk, ϕ). The
linear predictor ηijk that relates the mean to the link function is given by
ηijkl = θ þ αi þ βj þ ðαβÞij þ αβðr ÞijðlÞ þ τk þ ðατÞik þ ðβτÞjk þ ðαβτÞijk
Table 9.40 Percentage of
CO2 by bacterial activity as a
function of soil density
(mg soil/m3) and soil humidity
(kg water/kg soil)
Density
1.1
Humidity
0.1
0.2
0.24
1.4
0.1
0.2
0.24
1.6
0.1
0.2
0.24
Replication
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
Day 1
2.7
2.9
5.2
3.6
4
4.1
2.6
2.2
4.3
3.9
1.9
3
2
3
3.8
2.6
1.3
0.5
Day 2
0.34
1.57
5.04
3.92
3.47
3.47
1.12
0.78
3.36
2.91
3.02
3.81
0.67
0.78
2.8
3.14
2.69
0.34
Day 3
0.11
1.25
3.7
2.69
3.47
2.46
0.9
0.34
3.02
2.35
2.58
2.69
0.22
0.22
2.02
2.46
2.46
.
408
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.41 Analysis of variance of an CRD with factorial structure of treatments in repeated
measures
Sources of variation
Treatment
Humidity
Treatment*humidity
Error1
Time
Treatment time
Humidity*time
Treat*hum*time
Error2
Total
Degrees of freedom
(a - 1) = 3 - 1 = 2
(b - 1) = 3 - 1 = 2
(a - 1)(b - 1) = 4
ab(r - 1) = 3 × 3 × 1 = 9
(c - 1) = 3 - 1 = 2
(a - 1)(c - 1) = 4
(b - 1)(c - 1) = 4
(a - 1)(b - 1)(c - 1) = 8
/diferencia/17
a × b × c × r - 1 = 3 × 3 × 3 × 2 - 1 - 1 = 52
Note: Here, 1 degree of freedom was subtracted from the total observations of the experiment since
there is a missing observation
i = 1, 2, 3, j = 1, 2, 3, k = 1, 2, 3, l = 1, 2
where θ is the intercept, αi is the fixed effect of the density factor, βj is the fixed effect
of the humidity factor, (αβ)ij is the effect of the interaction between density and
humidity, αβ(r)ij(l ) is the random effect of the interaction density × humidity ×
repetition αβðr ÞijðlÞ N 0, σ 2αβðrÞ , τl is the fixed effect of measurement time,
(ατ)ij is the fixed effect of the interaction between density and measurement time,
(βτ)jk is the fixed effect of the interaction between moisture and measurement time,
and (αβτ)ijk is the fixed effect of the interaction of density × humidity × time. The
link function is defined by logit(π ijkl) = ηijkl.
The following SAS GLIMMIX syntax fits a repeated measures GLMM with a
beta distribution.
proc glimmix data=co2_fact nobound method=laplace;
class density moisture rep time;
model pct = density|humidity|time/dist=beta link=logit;
random density*humidity/subject=rep type=toep(1);
lsmeans density|humidity|time/lines ilink;
run;
Part of the results is listed below. The fit statistics (AIC and AICC) in Table 9.42
part (a) indicate that a Toeplitz covariance structure of order 1 provides the best fit to
of the data.
The type III tests of fixed effects in Table 9.43 indicate that soil density
(P = 0.0021), humidity (P = 0.0001), the evolution of emission over time
(P = 0.0001), and the interaction between moisture and time of measurement
(P = 0.0001) are statistically significant.
9.10
Joint Model for Binary and Poisson Data
409
Table 9.42 Fit statistics of a beta GLMM with a factorial structure of treatments under different
covariance structures
(a) Fit statistics
CS
-2 Log likelihood
-413.74
AIC (smaller is better)
-353.74
AICC (smaller is better)
-269.19
BIC (smaller is better)
-392.94
CAIC (smaller is better)
-362.94
HQIC (smaller is better)
-435.73
(b) Fit statistics for conditional distribution
-2 Log L (y | r. effects)
Pearson’s chi-square
Pearson’s chi-square/DF
AR(1)
Toep(1)
-413.72
-413.72
-353.72
-355.72
-269.18
-280.07
-392.93
-393.62
-362.93
-364.62
-435.71
-434.98
CS
AR(1)
Toep(1)
-413.74 -413.72 -413.72
64.60
64.65
64.65
1.22
1.22
1.22
UN
No converge
UN
No converge
Table 9.43 Hypothesis testing of the factors under study
Type III tests of fixed effects
Effect
Density
Humidity
Density*humidity
Time
Density*time
Humidity*time
Density*humidity*time
Num DF
2
2
4
2
4
4
8
Den DF
9
9
9
17
17
17
17
F-value
13.14
69.66
3.57
21.97
0.72
17.85
2.12
Pr > F
0.0021
<0.0001
0.0524
<0.0001
0.5904
<0.0001
0.0923
The least mean squares obtained with the “lsmeans” command on the model scale
are shown under the “Estimate” column and the data scale under the “Mean” column
of Table 9.44.
9.10
Joint Model for Binary and Poisson Data
Another advantage of the GLIMMIX procedure is the ability to fit models to data
where the distribution and/or link function varies with response variables. This is
accomplished through the specification of DIST = BYOBS or LINK=BYOBS in
the model definition. The dataset created below provides an example of a variable
with a bivariate outcome. This reflects the condition and length of hospital stay for
32 patients with herniorrhaphy. These data are taken from data provided by
Mosteller and Tukey (1977) and reproduced in the study by Hand et al. (1994)
(Table 9.45).
For each patient, two responses were recorded. A binary response takes the value
one if a patient experienced a routine recovery and the value zero if postoperative
intensive care was required. The second response variable is a count variable that
410
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.44 Means and standard errors and comparison of means (least significance difference
(LSD)) on the model scale (Estimate) and data scale (Mean)
(a) Density*humidity least squares means
Standard
Density Humidity Estimate error
DF t-value Pr > |t|
Mean
1.1
0.1
-4.5961 0.1614
9
-28.48 <0.0001 0.009990
1.1
0.2
-3.1829 0.07606
9
-41.85 <0.0001 0.03981
1.1
0.24
-3.3152 0.08060
9
-41.13 <0.0001 0.03505
1.4
0.1
-4.4567 0.1450
9
-30.74 <0.0001 0.01147
1.4
0.2
-3.3798 0.08333
9
-40.56 <0.0001 0.03293
1.4
0.24
-3.5363 0.08932
9
-39.59 <0.0001 0.02830
1.6
0.1
-4.7890 0.1809
9
-26.47 <0.0001 0.008252
1.6
0.2
-3.5453 0.08972
9
-39.52 <0.0001 0.02805
1.6
0.24
-4.3213 0.1440
9
-30.00 <0.0001 0.01311
(b) T grouping of density*humidity least squares means (α = 0.05)
LS means with the same letter are not significantly different
Density
Humidity
Estimate
1.1
0.20
-3.1829
1.1
0.24
-3.3152
B
1.4
0.20
-3.3798
B
1.4
0.24
-3.5363
B
1.6
0.20
-3.5453
B
1.6
0.24
-4.3213
1.4
0.10
-4.4567
1.1
0.10
-4.5961
1.6
0.10
-4.7890
Standard
error
mean
0.001596
0.002908
0.002726
0.001643
0.002654
0.002456
0.001481
0.002446
0.001863
A
A
A
C
C
C
C
measures the length of hospital stay after the surgery (in days). The binary variable
“OKstatus” is a regressor variable that distinguishes patients according to their
postoperative physical status (“1” implies better status), and the variable age is the
age of the patient.
These data can be modeled with a separate logistic model for the binary outcome
and with a Poisson model for the count outcome. Such separate analyses would not
take into account the correlation between the two response variables. It is reasonable
to assume that the duration of post-surgery hospitalization is correlated and will
depend on whether the patient requires intensive care.
In the following analysis, the correlation between the two types of response
variables for a patient is modeled with shared random effects (G-side). The dataset
variable “dist” identifies the distribution for each observation. For those observations
that follow a binary distribution, the response variable option “(event = “1 “)”
determines which value of the binary variable is modeled as the event of interest.
Since no “link” option is specified, the link is also chosen on an observation-byobservation basis as a predetermined link for the respective distribution. The following GLIMMIX commands fit this dataset with two distributions:
9.10
Joint Model for Binary and Poisson Data
411
Table 9.45 Hospital condition and length of stay of patients
D
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
Patient
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
16
16
Age
78
78
60
60
68
68
62
62
76
76
76
76
64
64
74
74
68
68
79
79
80
80
48
48
35
35
58
58
40
40
19
19
OKstatus
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
0
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
y
0
9
0
4
1
7
1
35
0
9
1
7
1
5
1
16
1
7
0
11
1
4
1
9
1
2
1
4
1
3
1
4
D
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
B
P
Patient
17
17
18
18
19
19
20
20
21
21
22
22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
30
30
31
31
32
32
Age
79
79
51
51
57
57
51
51
48
48
48
48
66
66
71
71
75
75
2
2
65
65
42
42
54
54
43
43
4
4
52
52
data Poi_Bin;
length dist $7;
input d$ patient age OKstatus response @@;
if d = 'B' then dist='Binary'; else dist='Poisson';
datalines;
B 1 78 1 0 P 1 78 1 9 B 2 60 1 0 P 2 60 1 4
B 3 68 1 1 1 P 3 68 1 7 B 4 62 0 1 P 4 62 0 35
.................................
......................................
...................................
OKstatus
0
0
1
1
1
1
0
0
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
y
0
3
1
5
1
8
1
8
1
3
1
5
1
8
0
2
0
7
1
0
0
16
0
3
0
2
1
3
1
3
1
8
412
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.46 Model information
Model information
Dataset
Response variable
Response distribution
Link function
Variance function
Variance matrix blocked by
Estimation technique
Degrees of freedom method
WORK.POI_BIN
Response
Multivariate
Multiple
Default
Patient
Residual pseudo-likelihood (PL)
Containment
....................................
.....................................
..................................
B 29 54 1 0 P 29 54 1 2 B 30 43 1 1 1 P 30 43 1 3
B 31 4 1 1 1 P 31 4 1 3 B 32 52 1 1 1 P 32 52 1 8
;
proc glimmix data=joint;
class patient dist;
model response(event='1') = dist dist*age dist*OKstatus /
noint s dist=byobs(dist);
random int / subject=patient;
lsmeans dist/lines ilink;
run;
Some of the output is shown below. Table 9.46 (“Model information”) shows that
the distribution of the data is multivariate and that possibly multiple link functions
are involved; by default, proc. GLIMMIX uses a logit link for the binary observations and a log link for the Poisson data.
Table 9.47 shows the value of the distribution statistic Gener. chi - square/
DF = 0.90, which indicates that there is no overdispersion, and also shows the
estimated variance component due to patient, which is, σ 2patient = 0:299. The fixed
effects tests for the effects of age and status are shown in part (c).
In addition to the above results, the maximum likelihood estimators of the
intercepts, as well as the values of the slopes of each of the variables of both
probability distributions, are tabulated in Table 9.48.
Thus, to calculate the probability that a patient will experience a routine recovery,
the following expression is used:
π^ =
=
1
1 þ exp
- β0 - β1 × age - β2 × okstatus
1
1 þ exp f - 5:7783þ0:07572 × ageþ0:4697 × okstatusg
9.11
Exercises
Table 9.47 Results of the
analysis of variance
413
(a) Fit statistics
-2 Res log pseudo-likelihood
Generalized chi-square
Gener. chi-square/DF
(b) Covariance parameter estimates
Cov Parm
Subject
Estimate
Intercept
Patient
0.2990
(c) Type III tests of fixed effects
Effect
Num DF
Den DF
Dist
2
29
Age*dist
2
29
OKstatus*dist
2
29
226.71
52.25
0.90
Standard error
0.1116
F-value
2.74
5.94
0.24
Pr > F
0.0814
0.0069
0.7909
t-value
1.99
1.48
-2.00
2.54
-0.42
-0.61
Pr > |t|
0.0562
0.1506
0.0552
0.0167
0.6794
0.5435
Table 9.48 Maximum likelihood estimators for fixed effects
Solutions for fixed effects
Effect
Dist
Dist
Binary
Dist
Poisson
Age*dist
Binary
Age*dist
Poisson
OKstatus*dist
Binary
OKstatus*dist
Poisson
Estimate
5.7783
0.8410
-0.07572
0.01875
-0.4697
-0.1856
Standard error
2.9048
0.5696
0.03791
0.007383
1.1251
0.3020
DF
29
29
29
29
29
29
whereas the following expression is used to calculate the average value of the length
of hospital stay after the surgery (in days):
^λ = exp fα0 þα1 × ageþα2 × okstatusg = exp f0:8410þ0:01875 × age - 0:1856 × okstatusg
9.11
Exercises
Exercise 9.11.1 Consider an experiment in which three treatments are compared.
There are r blocks of n animals, each using grouping criteria relevant to the
experiment. Within each block, one animal is randomly assigned to each treatment.
A measurement was taken on animals at “week 0,” when treatments were applied,
and again at weeks 4 and 12. Variables measured included weight, the presence or
absence of disease symptoms, and severity of symptoms, classified as “worse,” “no
change,” or “better.” The focus of this experiment was on repeated measures analysis
of the last two types of data in the above list: categorical data that are binary or
ordinal and ordinal responses/ratings in an experiment designed with a repeated
measures and treatment factor structure. Regardless of whether the observations are
414
9
Generalized Linear Mixed Models for Repeated Measurements
Table 9.49 Results of a repeated measures experiment with an ordinal response variable
Response
Bad
Without
change
Better
Week 0
Placebo
60
7
0
Trt1
59
6
Trt2
54
13
0
0
Week 4
Response
14
34
Placebo
5
33
Trt1
3
38
Week 12
Trt2 Response
13
10
25
17
Placebo
7
21
15
22
17
17
21
28
normally distributed, categorical, or have some other distribution, a general
approach to repeated measures analysis based on the linear mixed model uses the
following general form:
Observation = systematic between - subjects variation þ random between
- subjects variation þ systematic within - subjects effects þ random within
- subjects variation:
The following table shows the data from an experiment in which each cell
contains the number of animals in a given treatment × week × response category
combination (Table 9.49).
(a) List all the components of the repeated measures under a multinomial GLMM.
(b) Study and choose the best covariance structure that models this dataset. Cite the
most relevant results.
(c) Fit the multinomial cumulative logit model to these data. Perform a complete and
appropriate analysis of the data, focusing on:
(i) An evaluation of the effects of the combination of treatments
(ii) Odds ratio interpretation
(iii) The expected probability per category for each treatment
(d) Test whether the proportional odds assumption is viable. Cite relevant evidence
to support your conclusion regarding the adequacy of the assumption.
Repeat (b) through (d), assuming a generalized multinomial logit in Exercise
9.11.1. Discuss your results.
Repeat (b) through (d) assuming a multinomial cumulative probit in Exercise
9.11.1. Discuss your results and compare with those found in (1) and (2).
Alternatively, the contingency table approach can be implemented using a
log-linear model. For the previous example, 9.11.1, fit the log-linear model
log λijk = μ þ τi þ ϖ j þ ðτϖ Þij þ ck þ ðτcÞik þ ðτϖcÞijk
where λijk is the expected count of the treatment combination ijk by week by response
category and τ, ϖ, and c refer to treatment, week,and response category effects,
respectively.
9.11
Exercises
415
Table 9.50 Nitrogen injection treatment factors study
Handling practices
Application rate
1 = N surface applied without additional water injection
2.5 g/cm2
2 = N surface area applied with supplementary water injection
3 = N injected with a number 56 nozzle (7.6 cm depth of injection)
5.0 g/cm2
4 = N injected with a number 53 nozzle (12.7 cm depth of injection)
Handling practices 1
Driving practice 2
N2
Total
Quality
N1
N2
Total
Quality
N1
Poor
14
5
19
Poor
15
8
23
Average
2
11
13
Average
1
8
9
Good
0
0
0
Good
0
0
0
Excellent
0
0
0
Excellent
0
0
0
Total
16
16
32
Total
16
16
32
Handling practices 3
Handling practices 4
N2
Total
Quality
N1
N2
Total
Quality
N1
Poor
0
0
0
Poor
1
0
1
Average
9
2
11
Average
12
4
16
Good
7
14
21
Good
0
0
0
Excellent
0
0
0
Excellent
0
0
0
Total
16
16
32
Total
16
16
32
Exercise 9.11.2 Fertilization of turf has traditionally been accomplished through
surface applications. The introduction of new equipment (Hydroject) has made it
possible to place soluble materials below the surface (Table 9.50).
A study was conducted during the 1997 growing season to compare surface
application and subsoil injection of nitrogen on the green color of bentgrass
(Agrostis palustris L. Huds) 1 year after transplanting. The treatment structure was
a full factorial of grass management factors (four types/levels) and the rate/level (two
levels) of nitrogen application per square meter (g/m2). Eight treatment combinations were arranged in a completely randomized design with four replications. Turf
color was evaluated in each experimental unit at weekly intervals of 4 weeks as poor,
average, good, or excellent.
Of particular interest was the determination of the water injection effect, the
subsurface effect, and the comparison of injection versus surface applications. These
are contrasts between the levels of factor management practice and their primary
objective, which was to determine whether the factor interacts with the rate of
application.
(a) List all the GLMM components of this experiment.
(b) Fit the multinomial cumulative logit proportional odds model to these data.
Perform a complete and appropriate analysis of the data, focusing on:
(i) An evaluation of the effects of the combination of treatments
(ii) Interpretation of the odds ratios
(iii) The expected probability per category for each treatment
416
9
Generalized Linear Mixed Models for Repeated Measurements
(c) Test whether the proportional odds assumption is viable. Cite relevant evidence
to support your conclusion regarding the adequacy of the assumption.
Exercise 9.11.3 Refer to Exercise 9.11.1.
(a) Fit the multinomial generalized logit proportional odds model to these data.
(b) List all the components of the GLMM of this experiment.
(c) Perform a complete and appropriate analysis of the data, focusing on:
(i) An evaluation of the effects of the combination of treatments
(ii) Interpretation of odds ratios
(iii) The expected probability per category for each treatment
(d) Test whether the proportional odds assumption is viable. Cite relevant evidence
to support your conclusion regarding the adequacy of the assumption.
Exercise 9.11.4 Refer to Exercise 9.11.1.
(a) List all the components of the GLMM of this experiment.
(b) Fit the multinomial cumulative probit proportional odds model to these data.
Perform a complete and appropriate analysis of the data, focusing on:
(i) An evaluation of the effects of the combination of treatments
(ii) Interpretation of the odds ratios
(iii) The expected probability per category for each treatment
(c) Test whether the proportional odds assumption is viable. Cite relevant evidence
to support your conclusion regarding the adequacy of the assumption.
Appendix
Data: Feeding line experiment
Tray
Feeding
Run
1
H
1
2
H
1
3
H
1
4
H
1
5
H
1
6
H
1
7
H
1
8
H
1
9
H
1
10
H
1
11
H
1
12
H
1
Proportion
0.18217
0.15493
0.15906
0.15869
0.14891
0.17654
0.12915
0.12895
0.16688
0.11965
0.21719
0.20797
Tray
1
2
3
4
5
6
7
8
9
10
11
12
Feeding
H
H
H
H
H
H
H
H
H
H
H
H
Run
3
3
3
3
3
3
3
3
3
3
3
3
Proportion
0.06818
0.05874
0.05757
0.10349
0.08564
0.09359
0.09706
0.13188
0.18477
0.10966
0.18069
0.18182
(continued)
Appendix
Data: Feeding line experiment
Tray
Feeding
Run
1
L
1
2
L
1
3
L
1
4
L
1
5
L
1
6
L
1
7
L
1
8
L
1
9
L
1
10
L
1
11
L
1
12
L
1
1
H
2
2
H
2
3
H
2
4
H
2
5
H
2
6
H
2
7
H
2
8
H
2
9
H
2
10
H
2
11
H
2
12
H
2
1
L
2
2
L
2
3
L
2
4
L
2
5
L
2
6
L
2
7
L
2
8
L
2
2
9
L
10
L
2
11
L
2
12
L
2
417
Proportion
0.70601
0.68817
0.68317
0.77805
0.76692
0.79127
0.73653
0.74939
0.78773
0.7381
0.88486
0.90401
0.07547
0.05801
0.0565
0.09579
0.10954
0.12154
0.1144
0.13728
0.15012
0.12113
0.17633
0.16408
0.78318
0.78418
0.78589
0.78867
0.81988
0.82793
0.81384
0.81037
0.77528
0.78916
0.87109
0.84704
Tray
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
Feeding
L
L
L
L
L
L
L
L
L
L
L
L
H
H
H
H
H
H
H
H
H
H
H
H
L
L
L
L
L
L
L
L
L
L
L
L
Run
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
Proportion
0.75524
0.77249
.
0.84204
0.81572
0.79161
0.81234
0.81795
0.8225
0.79384
0.8135
0.83965
0.07105
0.05511
0.05217
0.10567
0.0755
0.0853
0.09363
0.11154
0.16264
0.09215
0.1834
0.21016
0.76556
0.78307
0.76486
0.82391
0.8044
0.81178
0.84339
0.78833
0.79804
0.82236
0.83807
0.85532
Data: Nitrous oxide emission
A
T
T
F
pH
3
1
1
108
8
2
2
1
15
4.7
1
3
1
33
5.5
4
4
1
23
6.4
3
1
2
-58
8
2
2
2
-45
4.7
1
3
2
-97
5.5
4
4
2
185
6.4
3
1
3
47
8
2
2
3
26
4.7
1
3
3
41
5.5
19
6.4
4
4
3
3
1
4
311
8
2
2
4
-29
4.7
1
3
4
37
5.5
4
4
4
204
6.4
3
1
5
6.8
8
2
2
5
-18
4.7
1
3
5
68
5.5
4
4
5
91
6.4
3
1
1
135
4.6
2
2
1
1.4
4.3
1
3
1
55
6.5
4
4
1
18
6.1
3
1
5
5
4.6
HE
85
86
85
84
85
86
85
84
85
86
85
84
85
86
85
84
85
86
85
84
84
85
85
85
84
TE
20
20
20
21
23
24
29
28
32
32
34
34
28
27
28
27
22
22
22
22
20
20
20
21
22
tx
17
17
17
17
34
34
34
34
35
35
35
35
30
30
30
30
27
27
27
27
18
18
18
18
24
tm
16
16
16
16
34
34
34
34
35
35
35
35
30
30
30
30
27
27
27
27
18
18
18
18
24
Hx
69
69
69
69
54
54
54
54
38
38
38
38
40
40
40
40
51
51
51
51
60
60
60
60
61
Hm
68
68
68
68
51
51
51
51
30
30
30
30
38
38
38
38
43
43
43
43
59
59
59
59
59
A
2
3
4
1
3
4
1
3
4
1
3
4
1
3
4
1
3
4
1
3
4
1
3
4
1
T
1
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
t
1
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
1
1
1
5
5
5
1
1
1
4.13
82.4
51.4
704
14
130
537
1.02
41.9
824
53.9
9.88
745
92.7
187
591
7
8.39
1367
-49
-91
711
108
15
621
F
pH
6.9
7
6.4
95
7
6.4
95
7
6.4
95
7
6.4
95
7
6.4
95
6.6
6.8
86
6.6
6.8
86
7.1
6.8
86
HE
87
87
87
20
87
87
24
87
87
24
87
87
20
87
87
20
85
87
20
85
87
19
86
87
20
TE
20
22
20
19
22
24
27
25
24
28
20
20
22
20
20
22
20
20
20
19
19
18
20
20
17
tx
19
19
19
19
27
27
27
28
28
27
22
22
22
22
22
22
20
20
18
18
18
18
17
17
16
tm
19
19
19
67
27
27
68
27
27
63
22
22
61
22
22
65
18
18
80
18
18
89
16
16
87
Hx
67
67
67
66
68
68
67
63
63
57
61
61
60
65
65
62
80
80
74
89
89
89
87
87
87
Hm
66
66
66
170
67
67
170
57
57
170
60
60
170
62
62
170
74
74
170
89
89
170
87
87
170
418
9
Generalized Linear Mixed Models for Repeated Measurements
2
1
4
3
2
1
4
3
2
1
4
3
2
1
4
3
2
1
4
1
3
2
1
3
2
1
3
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
3
4
1
3
4
1
3
5
5
5
1
1
1
1
5
5
5
5
1
1
1
1
5
5
5
5
1
1
1
2
2
2
3
3
12
121
51
21
87
21
28
31
101
-37
136
26
16
92
-82
35
-10
41
19
16
83
28
24
29
9.4
29
38
4.3
6.5
6.1
4.3
4.6
6.5
6.1
4.3
4.6
6.5
6.1
4.4
4.8
5.5
6
4.4
4.8
5.5
6
54
5.8
7.2
55
5.8
7.2
44
5.8
85
85
85
85
86
85
85
85
86
85
85
84
85
87
85
84
85
87
85
160
85
84
160
85
84
160
85
22
22
23
19
19
19
19
23
23
23
23
19
19
19
19
22
25
23
23
161
21
20
161
23
22
161
31
24
24
24
18
18
18
18
25
25
25
25
19
19
19
19
24
24
24
24
4
16
16
4
24
24
4
29
24
24
24
17
17
17
17
25
25
25
25
19
19
19
19
22
22
22
22
2
19
19
2
23
23
2
25
61
61
61
61
61
61
61
57
57
57
57
61
61
61
61
67
67
67
67
2
50
50
2
58
58
2
51
59
59
59
58
58
58
58
55
55
55
55
60
60
60
60
62
62
62
62
1
54
54
2
55
55
3
44
3
4
1
3
4
1
3
4
1
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
2
3
4
2
3
4
2
3
4
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
5
5
5
1
1
1
5
5
5
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
6.19
1.36
656
18.2
55.9
731
-77
33.6
880
29.4
49.3
69.7
36.1
-100
123
45.3
18.2
-216
-71
3.47
57.6
59.1
38.2
74.4
26
67.5
-77
7.1
6.8
86
7.3
7
90
7.3
7
90
7.5
7.1
6.7
6.8
7.5
7.1
6.7
6.8
7.5
7.1
6.7
6.8
7.5
7.1
6.7
6.8
7.5
7.1
86
87
25
86
87
19
86
87
26
88
88
85
90
88
88
85
90
88
88
85
90
88
88
85
90
88
88
26
24
24
20
20
18
26
25
25
20
20
20
24
24
24
24
30
30
30
30
26
26
26
26
24
24
24
24
24
24
18
18
17
25
25
25
20
20
20
42
42
42
42
39
39
39
39
31
31
31
31
26
26
26
24
24
65
17
17
77
25
25
65
21
21
21
29
29
29
29
34
34
34
34
28
28
28
28
25
25
25
64
64
170
76
76
170
63
63
163
59
59
59
20
20
20
20
29
29
29
29
23
23
23
23
31
31
31
(continued)
65
65
64
77
77
76
65
65
63
60
60
60
51
51
51
51
45
45
45
45
26
26
26
26
43
43
43
Appendix
419
Data: Nitrous oxide emission
A
T
T
F
pH
2
4
3
1.2
7.2
1
1
4
28
35
3
3
4
81
5.8
2
4
4
17
7.2
1
1
5
26
39
3
3
5
99
5.8
2
4
5
31
7.2
1
1
1
20
50
3
3
1
-45
6.4
2
4
1
108
7.4
1
1
5
27
47
2.3
6.4
3
3
5
2
4
5
39
7.4
1
1
1
24
53
3
3
1
13
6.4
2
4
1
148
7.1
1
1
5
24
57
3
3
5
9
6.4
2
4
5
13
7.1
1
1
1
22
69
3
3
1
-36
6.5
2
4
1
45
6.8
1
1
5
23
70
3
3
5
35
6.5
2
4
5
-17
6.8
HE
84
160
85
84
160
85
84
160
85
84
160
85
84
160
85
84
160
85
84
160
83
84
160
83
84
TE
31
161
27
29
161
20
20
161
20
22
161
28
28
161
20
20
161
28
28
161
20
20
161
22
22
tx
29
4
28
28
4
26
26
4
20
20
4
27
27
4
24
24
4
24
24
4
22
22
4
23
23
tm
25
2
27
27
2
26
26
2
20
20
2
26
26
2
22
22
2
23
23
2
21
21
2
22
22
Hx
51
2
35
35
2
41
41
2
54
54
2
50
50
2
54
54
2
60
60
2
73
73
2
71
71
Hm
44
4
35
35
5
39
39
1
50
50
5
47
47
1
53
53
5
57
57
1
69
69
5
70
70
A
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
T
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
t
5
1
1
1
1
5
5
5
5
1
1
1
1
5
5
5
5
1
1
1
1
5
5
5
5
40.8
37.5
88.4
63.4
-72
-83
95.1
72.8
12.2
27
23.2
-0.7
61.1
73.8
62.1
-96
40.2
7.22
5.53
74.5
-23
90.3
-21
63.9
-16
F
pH
6.7
7.2
8.1
7.6
7.4
7.2
8.1
7.6
7.4
6.9
8.1
7.4
7.3
6.9
8.1
7.4
7.3
7.3
7.9
7.4
7.4
7.3
7.9
7.4
7.4
HE
85
89
88
90
85
89
88
90
85
88
90
88
86
88
90
88
86
90
88
86
82
90
88
86
82
TE
24
22
21
22
22
26
26
26
26
21
22
22
24
19
20
20
20
17
17
18
19
20
20
20
20
tx
26
18
18
18
18
29
30
31
32
23
23
23
23
18
18
18
18
17
17
17
17
21
21
21
21
tm
25
17
17
17
17
27
28
29
30
22
22
22
22
18
18
18
18
17
17
17
17
20
20
20
20
Hx
43
61
61
61
61
48
49
50
51
63
63
63
63
76
76
76
76
73
73
73
73
70
70
70
70
Hm
31
48
48
48
48
45
46
47
48
58
58
58
58
65
65
65
65
67
67
67
67
69
69
69
69
420
9
Generalized Linear Mixed Models for Repeated Measurements
Appendix
421
Data: Percentage inhibition (Bio bioassay, Con concentration, Rep repetition, Por percentage
inhibition)
Bio
Day
Con
Rep
Por
Bio
Day
Con
Rep
Por
1
1
0
3
5.2632
1
6
2000
4
35.1724
1
1
0
4
5.2632
2
1
0
2
0.0016
1
1
500
1
15.7895
2
1
0
3
14.2857
1
1
500
2
26.3158
2
1
500
1
42.8571
1
1
500
3
15.7895
2
1
500
2
42.8571
1
1
500
4
15.7895
2
1
500
3
42.8571
1
1
1000
1
36.8421
2
1
500
4
42.8571
1
1
1000
2
36.8421
2
1
1000
1
7.1429
1
1
1000
3
36.8421
2
1
1000
2
42.8571
1
1
1000
4
36.8421
2
1
1000
3
42.8571
1
1
2000
1
15.7895
2
1
1000
4
42.8571
1
1
2000
2
36.8421
2
2
0
1
1.3699
1
1
2000
3
36.8421
2
2
0
2
1.3699
1
1
2000
4
36.8421
2
2
0
4
1.3699
1
2
0
2
1.9355
2
2
500
1
34.2466
1
2
0
3
4.5161
2
2
500
2
31.5068
1
2
0
4
1.9355
2
2
500
3
42.4658
1
2
500
1
43.2258
2
2
500
4
36.9863
1
2
500
2
48.3871
2
2
1000
1
34.2466
1
2
500
3
40.6452
2
2
1000
2
47.9452
1
2
500
4
40.6452
2
2
1000
3
45.2055
1
2
1000
1
35.4839
2
2
1000
4
45.2055
1
2
1000
2
45.8065
2
2
2000
1
47.9452
1
2
1000
3
43.2258
2
2
2000
2
53.4247
1
2
1000
4
32.9032
2
2
2000
3
50.6849
2000
4
56.1644
1
2
2000
1
58.7097
2
2
1
2
2000
2
53.5484
2
3
0
1
4.2735
1
2
2000
3
53.5484
2
3
0
4
14.5299
1
2
2000
4
58.7097
2
3
500
1
28.2051
1
3
0
2
1.2346
2
3
500
2
28.2051
1
3
0
3
3.7037
2
3
500
3
35.0427
1
3
500
1
25.9259
2
3
500
4
24.7863
1
3
500
2
23.4568
2
3
1000
1
24.7863
1
3
500
3
23.4568
2
3
1000
2
35.0427
1
3
500
4
24.6914
2
3
1000
3
24.7863
1
3
1000
1
30.8642
2
3
1000
4
26.4957
1
3
1000
2
32.0988
2
3
2000
1
40.1709
1
3
1000
3
28.3951
2
3
2000
2
38.4615
1
3
1000
4
25.9259
2
3
2000
3
47.0085
1
3
2000
1
53.0864
2
3
2000
4
41.8803
1
3
2000
2
49.3827
2
4
0
2
1.5015
1
3
2000
3
49.3827
2
4
0
3
1.5015
(continued)
422
9 Generalized Linear Mixed Models for Repeated Measurements
Data: Percentage inhibition (Bio bioassay, Con concentration, Rep repetition, Por percentage
inhibition)
Bio
Day
Con
Rep
Por
Bio
Day
Con
Rep
Por
1
3
2000
4
51.8519
2
4
0
4
1.5015
1
4
0
3
4.6729
2
4
500
1
20.7207
1
4
500
1
19.6262
2
4
500
2
23.1231
1
4
500
2
20.5607
2
4
500
3
27.9279
1
4
500
3
22.4299
2
4
500
4
20.7207
1
4
500
4
20.5607
2
4
1000
1
35.1351
1
4
1000
1
21.4953
2
4
1000
2
26.7267
1
4
1000
2
21.4953
2
4
1000
3
26.7267
1
4
1000
3
23.3645
2
4
1000
4
32.7327
1
4
1000
4
20.5607
2
4
2000
1
33.9339
1
4
2000
1
42.0561
2
4
2000
2
37.5375
1
4
2000
2
36.4486
2
4
2000
3
44.7447
3
32.7103
2
4
2000
4
38.7387
1
4
2000
1
4
2000
4
40.1869
2
5
0
2
2.008
1
5
0
3
4.065
2
5
0
4
0.4016
1
5
0
4
4.065
2
5
500
1
13.253
1
5
500
1
21.1382
2
5
500
2
21.2851
1
5
500
2
24.3902
2
5
500
3
21.2851
1
5
500
3
17.0732
2
5
500
4
18.0723
1
5
500
4
17.0732
2
5
1000
1
21.2851
1
5
1000
1
18.6992
2
5
1000
2
18.0723
1
5
1000
2
18.6992
2
5
1000
3
16.4659
1
5
1000
3
20.3252
2
5
1000
4
16.4659
1
5
1000
4
17.8862
2
5
2000
1
35.743
1
5
2000
1
41.4634
2
5
2000
2
34.1365
2000
3
29.3173
1
5
2000
2
38.2114
2
5
1
5
2000
3
34.1463
2
5
2000
4
30.9237
1
5
2000
4
33.3333
2
6
0
2
4.2159
1
6
0
3
4.8276
2
6
0
4
0.1686
1
6
0
4
2.069
2
6
500
1
18.3811
1
6
500
1
17.2414
2
6
500
2
20.4047
1
6
500
2
18.6207
2
6
500
3
22.4283
1
6
500
3
16.5517
2
6
500
4
20.4047
1
6
500
4
13.7931
2
6
1000
1
21.0793
1
6
1000
1
15.8621
2
6
1000
2
17.7066
1
6
1000
2
16.5517
2
6
1000
3
17.7066
1
6
1000
3
15.8621
2
6
1000
4
20.4047
1
6
1000
4
18.6207
2
6
2000
1
31.1973
1
6
2000
1
32.4138
2
6
2000
2
29.1737
6
2000
2
29.6552
2
6
2000
3
29.8482
1
1
6
2000
3
31.7241
2
6
2000
4
30.5228
Appendix
423
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References
Agresti A (2013) Introduction to categorical datikea analysis, 3rd edn. Wiley, Hoboken
Aitchison J, Silvey S (1957) The generalization of probit analysis to the case of multiple responses.
Biometrika 44:131–140
Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In:
Petrov BN, Casake F (eds) Second international symposium on information theory. Akademiai
kiado, Budapest, pp 267–281
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Automat Control 19:
716–723
Amoah S, Wilkinson M, Dunwell J, King GJ (2008) Understanding the relationship between DNA
methylation and phenotypic plasticity in crop plants. Comp Biochem Physiol 150(Suppl 1):
S145
Bekele A, Bultosa G, Belete K (2012) The effect of germination time on malt quality of six sorghum
(sorghum bicolor) varieties grown at Melkassa, Ethiopia. J Brew 118(1):76–81
Bilgili S, Hess J, Blake J, Macklin K, Saenmahayak B, Sibley J (2009) Influence of bedding
material on footpad dermatitis in broiler chickens. J Appl Poult Res 18(3):583–589
Bliss CL (1934) Methods of probits. Science 79:38–39
Bliss CI (1935) The calculation of the dosage-mortality curve. Ann Appl Biol 22(1):134–167
Breslow NE (2004) Whither PQL? In: Lin DY, Heagerty PJ (eds) Proceedings of the second Seattle
symposium in biostatistics: analysis of correlated data. Springer, pp 1–22
Breslow NE, Clayton DG (1993) Approximate inference in generalized linear mixed models. J Am
Stat Assoc 88:9–25
Casella G, Berger RL (2002) Statistical inference, 2nd edn. Duxbury, Pacific Grove
Collett D (2002) Modelling binary data, 2nd edn. Chapman & Hall/CRC Press, Boca Raton. 387 pp.
De Jong IC, Guémené D (2012) Major welfare issues in broiler breeders. Worlds Poult Sci J 67:73–
82
Engel B, te Brake J (1993) Analysis of embryonic development with a model for under-or
overdispersion relative binomial variation. Biometrics 49:269–279
Fisher RA (1925) Statistical methods for research workers. Oliver and Boyd, Edinburgh
Garcia RG, Almeida PICL, Caldara FR, Naas IA, Pereira DF et al (2010) Effect of litter material on
water quality in broiler production. Braz J Poult Sci 12:165–169
Gbur EE, Stroup WW, McCarter KS, Durham S, Young LJ, Christman M, West M, Kramer M
(2012) Analysis of generalized linear mixed models in the agricultural and natural resources
sciences. ASA, CSSA, SSSA, Madison
Gilks WR et al (1996) Introducing Markov chain Monte Carlo. In: Gilks WR (ed) Markov chain
Monte Carlo in practice. Chapman and Hall, pp 1–19
© The Editor(s) (if applicable) and The Author(s) 2023
J. Salinas Ruíz et al., Generalized Linear Mixed Models with Applications
in Agriculture and Biology, https://doi.org/10.1007/978-3-031-32800-8
425
426
References
Goldstein H, Rasbash J (1996) Improved approximations for multilevel models with binary
responses. J R Stat Soc Ser A Stat Soc 159:505–513
Hand DJ, Daly F, Lunn AD, McConway KJ, Ostrowski E (1994) A handbook of small data sets.
Chapman and Hall, London
Heindel J, Price C, Field E, Marr M, Myers C, Morrissey R, Schwetz B (1992) Developmental
toxicity of boric acid in mice and rats. Toxicol Sci 18(2):266–277
Henderson CR (1950) Estimation of genetic parameters. Ann Math Stat 21:309–310
Henderson CR (1984) Applications of linear models in animal breeding. In Univ. of Guelph,
Guelph
Hosmer DW, Lemeshow S (2000) Applied logistic regression, 2nd edn. Wiley, New York
Immer RF, Hayes HK, Powers LR (1934) Statistical determination of barley varietal adaptation. J
Am Soc Agron 26:403–419
Jermann R, Toumiat M, Imfeld D (2001) Development of an in vitro efficacy test for self-tanning
formulations. Int J Cosmet Sci 24(1):35–42
Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2. Wiley,
New York
Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat
Assoc 53(282):457–481
Lee Y, Nelder JA (2001) Hierarchical generalised linear models: a synthesis of generalised linear
models, random-effect models and structured dispersions. Biometrika 88:987–1006
Lee Y, Nelder JA (2004) Conditional and marginal models: anhother view. Stat Inference 19(2):
219–238
Lew M (2007) Good statistical practice in pharmacology. Br J Pharmacol 152(3):299–303
Littell RC, Milliken GA, Stroup WW, Wolfinger RD (1996) SAS for mixed models. SAS Institute,
Inc., Cary
Littell RC et al (2006) SAS for Mixed Models, 2nd edn. SAS Publishing
Limpert E, Stahel WA, Abbt M (2001) Log-normal Distributions across the Sciences: Keys and
Clues. BioScience 51(5)
Logan M (2010) Biostatistical design and analysis using R: a practical guide. Wiley
Madden L, Hughes G (1995) Plant disease incidence: distributions, heterogeneity, and temporal
analysis. Annu Rev Phytopathol 33:529–564
Margolin BH, Kaplan N, Zeiger E (1981) Statistical analysis of the Ames Salmonella/microsome
test. Proc Natl Acad Sci USA 76:3779–3783
Martrenchar A, Boilletot E, Huonnic D, Pol F (2002) Risk factors for foot-pad dermatitis in chicken
and Turkey broilers in France. Prev Vet Med 52(3–4):213–226
McCullagh P (1980) Regression models for ordinal data. J R Stat Soc Series B Methodol 42:109–
142
McCullagh P (1983) Quasi-likelihood functions. Ann Stat 11(1):59–67
McCullagh P, Nelder JA (1989) Generalized linear models, 2nd edn. Chapman and Hall, London
Mead J, Curnow R, Hasted A (1993) Statistical methods in agriculture and experimental biology,
2nd edn. Chapman and Hall, London, p 325
Mosteller F, Tukey JW (1977) Data analysis and regression. Addison-Wesley, Reading
Myers R, Montgomery D, Vining G (2002) Generalized linear models with applications in
engineering and the sciences. Wiley, New York
Nadia Hernández-Tapia, Josafhat Salinas-Ruiz, Vinisa Saynes-Santillán, Julio M. AyalaRodríguez, Francisco Hernández-Rosas y Joel Velasco-Velasco (2019). N2O, CO2 and NH3
emission from dung of bovine with different percentage of crude protein in diet. Rev. Int.
Contam. Ambie 35 (3) 597–608. DOI: 10.20937/RICA.2019.35.03.07
Nelder JA, Wedderburn RWM (1972) Generalized linear models. J R Stat Soc Ser A 135:370–384
Pinheiro JC, Bates DM (2000) Mixed-effects models in S and SPLUS. Springer
Pinheiro JC, Chao EC (2006) Efficient Laplacian and adaptive Gaussian quadrature algorithms for
multilevel generalized linear mixed models. J Comput Graph Stat 15:58–81
References
427
Quinn GP, Keough MJ (2002) Experimental design and data analysis for biologists. Cambridge
University Press, New York
Raudenbush SW et al (2000) Maximum likelihood for generalized linear models with nested
random effects via high-order, multivariate Laplace approximation. J Comput Graph Stat 9:
141–157
Robinson GK (1991) That BLUP is a good thing: The estimation of random effects (with
discussion), Statist Sci 6:15–51
Rodriguez G, Goldman N (2001) Improved estimation procedures for multilevel models with
binary response: a case-study. J R Stat Soc Ser A Stat Soc 164:339–355
Schall R (1991) Estimation in generalized linear models with random effects. Biometrika 78:719–
727
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464
Shinozaki K, Kira T (1956) Intraspecific competition among higher plants. VII. Logistic theory of
the C-D effect. J Inst Polytech 12:69–82
Smith T, Mlambo V, Sikosana JLN, Maphosa V, Mueller-Harvey I, Owen E (2005) Dichrostachys
cinerea and Acacia nilotica fruits as dry season feed supplements for goats in a semi-arid
environment. Anim Feed Sci Technol 122(1–2):149–157
Spurgeon J (1978) The correlation of animal response data with the yields of selected thermal
decomposition products for typical aircraft interior materials. U.S. D.O.T. Report No. FAA-RD78-131
Stanley VG (1981) The effect of stocking density on commercial broiler performance. Poult Sci 60:
1737–1738
Stephens PA et al (2005) Information theory and hypothesis testing: a call for pluralism. J Appl Ecol
42:4–12
Stroup W (2012) Generalized linear mixed models, 1st edn. Chapman & Hall/CRC
Stroup W (2013) Generalized linear mixed models. CRC Press, Boca Raton
Taira K, Nagai T, Obi T, Takase K (2014) Effect of litter moisture on the development of footpad
dermatitis in broiler chickens. J Vet Med Sci 76:583–586
Wagner JG, Aghajanian GK, Bing OHL (1968) Correlation of performance test scores with "tissue
concentration" of lysergic acid diethylamide in human subjects. Clin Pharmacol Ther 9(5):
635–638
Wallsten TS, Budescu DV (1981) Adaptivity and nonadditivity in judging MMPI profiles. J Exp
Psychol Hum Percept Perform 7:1096–1109
Walters KJ, Hosfield GL, Uebersax MA, Kelly JD (1997) Navy bean canning quality: correlations,
heritability estimates and randomly amplified polymorphic DNA markers associated with
component traits. J Am Soc Hortic Sci 122(3):338–343
Wolfinger R, O'Connell M (1993) Generalized linear mixed models: a pseudo-likelihood approach.
J Stat Comput Simul 48:233–243
Yates F (1935) Complex experiments, with discussion. J R Stat Soc Ser B 2:181–223
Zeger SL, Liang K-Y, Albert PS (1988) Models for longitudinal data: a generalized estimating
equation approach. Biometrics 44(4):1049
Zuur AF, Leno EN, Walker N, Saveliev AA, Smith GM (2009) Mixed effects models and
extensions in ecology with R. Springer, New York
Zuur AF, Hilbe JM, Leno EN (2013) A Beginner's guide to GLM and GLMM with R: a frequentist.
and Bayesian perspective for ecologists. Highland Statistics Ltd., Newburgh